Using Computing Technologies to Enhance Mathematics Education
Dr. Robert P. King
April 23, 1998
Two billion instructions per second! Is that the speed of the human brain? No, that is the predicted speed of computer processors at the turn of the century. In 1965, Intel co-founder Gordon Moore predicted transistor density on microprocessors would double every two years (Intel). This prediction, so far, has proven amazingly accurate. If it continues, by the turn of the century, Intel processors should contain between 50 to 100 million transistors and execute the predicted two billion instructions per second.
Computing technology is always changing. And this computing technology has been introduced into the field of mathematics education. Years ago, a bulky Texas Instruments calculator was available. Not only could it add and subtract, but also could do multiplication as well as division. And the cost, just $99. Today, calculators, and even computers, are being used to teach everyone’s favorite subject, math!
Computing technology is slowly being integrated into standard curriculum. Should technology be used to learn math? Some feel that we lose the true meaning of the math and ideas. Others say that calculators give students an easy answer. Even in a recent Simpson’s episode, the teacher asked, "whose calculator has the answer to the problem?" If technology is used in this manner, then technology does not help. However, if used properly, technology can greatly increase a student’s understanding of the abstract mathematical ideas behind problems.
In the 1991 publication of the National Council of Teachers of Mathematics Commission on Teaching Standards, the use of computing technologies are recommended. Further, the commission suggested that teachers should accept and encourage the use of computers, calculators and other technologies. The Mathematical Association of America (1991) states that available technology has profoundly changed the teaching and learning of mathematics at all levels. Using such tools helps students to visualize purely abstract ideas resulting in a deeper understanding of the underlying concepts. Computing technologies such as function graphers, spreadsheets, and symbol manipulators also help students to pose problems, test conjectures, explore patterns, conduct simulations as well as organize and represent data. At Texas A&M University, the Department of Mathematics requires students in secondary mathematics education programs to complete a course in the integration of mathematics and technology (Schielack, 1997).
There are many forms of technology available to help students. In this paper, four different types of computing technology will be discussed with regards to the strengths and weaknesses of each. The first is the use of calculators such as the TI 80 series. The second is the use of computer software programs, which include Maple and Minitab. Third is the use of computers and CD-ROM, CD-W, and DVD discs to enhance classes. Last, the entity we call the Internet can be used to help teach math.
Calculators are everywhere. And in math, it is one of the most often used tools. There are several ranges of calculators available to students learning math. There are standard calculators that add, subtract, multiple, and divide. Some more advanced calculators handle fractions and trigonometric functions such as sine and tangent. However, graphing calculators are the biggest help in teaching mathematics at a higher level. These powerful machines can perform complex problem solving and provide graphical representations of two and even some three dimensional functions.
In some schools, mathematics education has adapted a new curriculum. However, it has been difficult to find a suitable curriculum for both educators and the public to agree on (Esty, 1998). In one case, most of the learning is done using graphing calculators. The key to teaching then revolves around abstract ideas, using examples that do not provide exact numbers or functions supplemented by examples on the calculator. One fundamental aspect of algebra is the concept domain and range. When using a graphing calculator to view classic graphs, these concepts can become very clear to the student. Esty found that calculators could facilitate the development of essential concepts of algebra in ways that were hardly possible before graphs were easy to draw and redraw. Even with calculators, tests can be based on the concepts using abstract questions and arbitrary graphs.
One of the most tedious sections of mathematics has been helped by the use of calculators: statistics. Statistics can often turn students off due to the large number of calculations needed. Through the use of a TI-82 calculator, the study of statistics has become more convenient for students (Crider, 1997). The calculator can easily store the data in a list. This data can be entered by hand or transferred over a Link cable connecting two calculators. Once the data is entered, the calculator can draw histograms of the data. Another useful tool in statistics is the use of box plots. This provides a detailed graphical representation as well as the ability to trace and find quartile breaks. In addition, the TI-82 is capable of producing three box plots simultaneously allowing for easy comparison.
Another powerful feature of a graphic calculator is the solver function. What is a solver? A solver is an utility that allows the entry of a function with any number of variables and after assignment of variable values, will numerically approximate the "unknown" variable (Dion, 1997). The solver utility is useful for both the teacher and the student. For a teacher, the solver is a tool that assists in the grading of individualized work. For students, a solver provides an easy way to explore mathematical concepts, build number sense, and problem solving abilities. In algebra, a solver can help build concepts. The student first must transform the problem into an algebraic expression. From there, all but one of the variable values must be entered. Finally, for speed, the student must provide an estimated value for the answer. The solver allows students to investigate problems without the burden of complicated computations or symbol manipulation.
In today’s information age, the privacy of data has been a growing concern. To students, the secretive nature of coding theory leads to much fascination in the mathematics of cryptology (St. John, 1998). In encryption, one method is the use of substitution, or changing one letter for another. Since certain letters occur more frequently, a strategy of substituting a group of letters for another makes breaking the code much harder. To encode groups of letter, a matrix and modulus arithmetic are used, both of which are handled by a graphics calculator. While the math behind this type of encryption is complex, the use of the calculator allows for students to understand the basic principles while the calculator performs the complex calculations.
While the calculator can be a valuable resource in mathematics, it can also cause many errors. Due primarily to architectural and numerical algorithmic limitations, graphing calculators can give surprising, misleading, and incorrect answers (Cruthrids & Dodd, 1997). This does not lessen the value of calculators in the classroom, however, teachers and students must be aware the results may not be correct and are not correct 100% of the time. The TI-85 calculator, just as all calculators and computers, perform finite arithmetic compared to infinite arithmetic. The TI-85 stores 14 digits of an answer and displays 12 of those. One example involves square roots. In theory, (Ö 2)2 = 2 and (Ö 3)2 = 3. In the TI-85, the first is correct but the second is not. The problem comes from a rounding error. The TI thinks of Ö 2 as 1.4142135623731 and exact square is 2.000 000 000 000 014 004 103 603 61 (spaces provided for easy reading) which the TI views as 2.000 000 000 000 0, or equal to 2. Even with fractions, 0.333 333 333 333 is not equal to 1/3. Now while this may appear obvious, how the calculator handles problems with a 1 in the numerator varies from that of a fraction, creating confusing "imaginary" results.
Graphing can also cause many problems. Cruthirds and Dodd (1997) also found problems when graphing piecewise functions. Piecewise functions are those that have different cases such as: when x is less than 0, return –x, and when x is greater than or equal to 0, return Ö x. In the calculator, this is represented by y = -x(x<0) + (Ö x)(x ³ 0). In theory, this function is correct. However, the calculator does not know how to handle the Ö x for negative numbers in "real" terms. Since the calculator returns an "imaginary" answer, it does not graph anything until x = 0 when it can fully solve the equation in "real" terms.
In using more advanced features, the calculator may not always return the answer needed because many solutions may exist for the data given. If a person has only a section of data and does not know what the data represents, errors can be made. Dobbs and Peterson (1997) found an interesting problem to study. An advanced feature, known as regression, attempts to develop a function based on the data provided. A table of data is provided that creates an arc like the top quarter of a circle. When regression is used, the calculator returns a parabolic equation. A parabolic equation is one in the form of ax2 + bx + c causing a similar shaped arc. With this answer, the problem looks like a rainbow going down forever. However, the data really represents the hours of daylight per a given day in a non-leap year in Washington, D.C. The amount of daylight per day is really cyclic or circular creating a wave pattern with peaks and valleys, creating a very different answer. With only part of the data, an overall wrong answer was provided. In addition, the calculator returned (-4.1157057228486*10-4) x2 + 0.14091126730412 x + 3.050425146283 as it’s answer. While correct, it is extremely precise and difficult to comprehend. One must be careful with exact, literal answers versus workable (i.e. rounded) answers.
The actual desktop computer has been integrated into many schools. Some of the more advanced schools even have computers in the classrooms. But computers are not just for word processing. The use of computers in mathematics has been felt in many higher levels of math for a long time. However, it has recently been used to teach high school math. Algebra represents a major turning point in a students mathematics career (Chamberlain, 1998). In teaching math with computers, the ability to teach at the student’s individual pace can be accomplished.
Several points must be considered when teaching math with computers. Specifically, when teaching algebra, most of the students are in 7th or 8th grade. With this age group, attention spans are short. In a recent Technology and Learning (Chamberlain, 1998) article, three of the most successful computer programs were analyzed. The best program, Astro Algebra, sets math in a futuristic setting on a space ship. The students play the part of the ship’s captain and may choose to "go on duty" and complete a series of "missions." While holding their attention, the students have access to several tools. These tools include a grapher, equivalence charts, and a three-dimensional block program that helps students visualize functions. There is also an "Astro Net," a set of definitions that can be referenced from anywhere in the program. The use of such programs can and does help students learn math. The computer allows the students to learn at their own pace as well as providing support and tools where necessary.
One application that has been used for years in the business world finally found it’s way to the teaching front: spreadsheets. The most popular spreadsheet program is Microsoft’s Excel. Spreadsheets are especially adaptable for problems whose conceptual format is tabular, iterative, or recursive (Aieta, 1997). There are several advantages to using a spreadsheet. For one, it allows the users to view the results of calculations as they change. In spreadsheets, many cells are directly related to other cells, such as totals. By simply changing one value, the whole page changes to adapt to the change. With this ease of calculation, the student is free from performing "messy" and time-consuming hand crunching. Also, the ability to easily see results promotes "what if" questioning and exploration. As mentioned previously, often times a cell is dependent upon other values. To create these cells the student is required to think of the algorithms to generate the correct result. Utilization of the software can be used extensively in a class like statistics or incidentally to generate graphs to help with visualization problems.
Spreadsheets are just not limited to tabular or data analysis. Spreadsheets can be successfully used in a standard mathematics class. Robert Iovinelli (1998) found gratifying results when he introduced computers into his secondary level classroom. Because of their design, spreadsheets are very useful for organizing information. Besides numbers, letters and other pieces of information can be used in the tables. Because of this ability, using a spreadsheet simply to organize and print data without performing calculations can be of great help. Again, the adaptability of spreadsheets to recalculate results encourages the use of "try and see" logic.
Another type of spreadsheet program is Minitab. Minitab is a combination of both a spreadsheet and a calculations program. One large advantage to this is in the field of statistics. Often in statistics, simulations are used to model real life data. The problem with simulation is that it requires a large number of repetitions. For problems with ½ probability, a fair coin is tossed. If tossed 10 times, the probability of five heads and five tails coming up are slim. However, if a coin is tossed 10,000 times, the odds will become closer to 50% for each side. Imagine tossing a coin 10,000 times, recording the results, and totaling them up. With Minitab, a standard computer can do all this in approximately five seconds. Other cases where the probability is 1/6 or 1/3, a die can be used for simulation. Again, the problem comes in repetition and the lack of ease in analyzing the large quantities of rolls. In using the computer to run the simulations, the results become more standard. These standard results create a "bell curve" plot with 94% within two standard deviations of the mean.
With the further use of computers in classrooms, there needs to be a way to store the data. Often, the use of computers involves large graphs, multimedia presentations, digital videos, and software programs with support files. These files are becoming so large that standard floppy disks cannot hold them. Lugo (1997) decided to try a new tactic to store the files: CD-ROM. CD-ROM stands for compact disc – read only memory. This uses a CD to store large volumes of data. A standard CD-ROM holds the equivalent of 450 disks. A new technology called DVD, or digital versital disc, looks similar to a CD but can hold even more data. A CD holds 650 megabytes or 0.650 gigabytes where as a DVD can hold 19 gigabytes, or about 13,200 standard disks. With present technologies, recording a CD-ROM has become a simple matter of transferring files from one disk to another. The rewards in terms of convenience and versatility of locally created disks can make the acquisition of a CD recorder for a school worthwhile.
In addition, a CD should be viewed as a resource, not a textbook. Some of the contents of the CD-ROM include materials that students can use. First, a series of presentations from classes are included. Next, a collection of programs written in a "visual" program such as Visual Basic, help students explore class problems. A laboratory manual, if one exists, is also included on the CD. Files from programs such as Minitab or Microsoft’s Excel that are used in classes are also available. If developed, a history of mathematics or a dictionary of definitions should be included. Multimedia files such as video, images, and audio, as well as any needed drivers to view the files should be included on the CD-ROM.
When using multimedia presentations, many aspects must be considered (Lugo, 1997). Sans serif fonts such as Arial with a minimum size of 14 points should be used. Teachers should also be careful when using True Type fonts. Often these fonts which are only local can cause problems when moved to another machine. Light background colors with a dark black text color show up the best when viewed from an audience. Contrary to what many teachers believe, lights can be on in a room while the presentation is running. The only consideration is to make sure that light is not shining directly on the screen. While presentation programs have the capability to be fully automated and timed, the teacher should use "click control" to allow students time to ask questions. Finally, many programs allow the teacher to print out copies of the presentation as well as allow room to write notes. This allows students to focus on the material rather than on taking notes on the content of the slides.
In another school, a CD-ROM is used in conjunction with World Wide Web (Beilby, 1998). Much of their materials are being converted to and distributed in an electronic format. This trend seems to be driven by consumers who are demanding materials for all subjects, not just mathematics. At this school, there are two components. The first is a collection of multimedia materials on CD-ROM. The second is a set of guidance texts on the web. The CD-ROM is based on a workbook and contains descriptive materials (course book), illustrative materials (lectures), worked exercises and examples (class demonstration), sticky notes (jottings and notes), and appendices (tutorials). The main advantage to the CD-ROM over conventional textbooks is that the system can cross-reference keywords in topics, pages, and chapters. The web section contains structured learning sheets (course handouts), tests, assignments, and assessment. The CD-ROM material is more static and takes longer to prepare while the web materials are variable and take different forms for different groups of students.
This use of the World Wide Web, or the Internet, is a technology that is still very young. The first use of the World Wide Web was in 1991 (Zakon). Since then, some upper level schools have begun teaching courses with the help of online references or completely in cyberspace. One school’s basic math course was offered in a standard format as well as an online format. The online course consists of thirty lessons, each with eight parts (Harris & Pfaff, 1998). The first part of each lesson was an overview, which provides students with a brief summary of the topics in the lesson. The second part is the lecture section which acts as a typed out in class lecture. Next are "expanded lectures," or worked out examples done in class. After the lecture sections, there are exercises that serve as homework or independent practice. Following this, there is an online quiz. An innovative section called "What’s It For?" attempts to answer why the student should know this section or material. A vocabulary section covers any new words that are used in this lesson. The final section is a writing assignment where the students write two to three paragraphs that synthesize the material learned in that section.
Harris and Pfaff (1998) found several advantages to an online course. For one, on the bottom of every page was a "Panic Button," a direct email link to the teacher in case there were any problems at any time. Another option added was a newsboard, or online bulletin board system. This allowed students to "talk" to each other and read results from other students. In the future, instantaneous chat rooms will be available and staffed during certain hours, or the equivalent of drop in hours. In addition, calculators and graphing utilities were also available online for student’s use. However, there were also some troubles with the online course. The hardest obstacle was how to get mathematical symbols onto web pages. Some signs are already on a keyboard, such as +, while others can be inferred, such as ^ to represent "raised to a power." Using a simple program written in Perl, other commands could be included in < >, just as commands normally are. The program simply returns a graphics file of the needed symbol. The final problem was with the authorization of students to take the quizzes. Each student had an user ID and password; however, anyone who knew it could log in as that person and take the quiz. While the quizzes were on "the honor system," final exams were taken "in person" with the teacher.
As mentioned in the previous section, displaying mathematical notation on the Internet can become very complicated. Three main strategies have emerged for displaying mathematical notations on the web (Miner & Scheftic, 1998). The first is to allow the browser to handle the code. This is limited to sub- and super-scripts, and some two dimensional items such as fractions. Other items can be made in other programs and saved as graphics. The second method is to use an external program that is designed to handle mathematical documents such as LaTeX or DVI. In this option, the files are simply available for download and are then opened in the special program. The third, which is becoming the most popular option, is a hybrid solution. In this case, the browser handles the management of the page while a plug-in or Java handles the mathematical notations. A plug-in is a mini program that helps a browser. In this case, the web page would have normal items as well as other parts that are displayed through Java or a plug-in. This provides easy cross-platform (IBM to Mac and vice versa) use as well as easy printing.
Using the hybrid idea, the problem of displaying mathematical notation on the web is helped with the use of Scientific Notebook (Gosselin & Yang, 1998). Scientific Notebook is a small version of Scientific Workplace, a program for writing mathematical and science textbooks, that plugs into Microsoft’s Internet Explorer. Scientific Notebook helps by allowing students to view documents without downloading the file. A system known as "natural language" is used to program in Scientific Notebook. Natural language is very similar to the pencil and paper version of mathematics. One added advantage to Scientific Notebook is that it allows students to easily create and post their own web pages with scientific notations.
With increasing demand for instant communication, web conferencing has become very popular. Computer conferencing is a system to support group communication (Harasim, 1993). This form of communication is useful when the members of a class are not all in the immediate area. These systems encourage collaborative work. The text based nature of the system forces students to engage in the discussion as well as to formulate ideas into written form. The system can also automatically archive transcripts of the sessions as well as generate reports that cover each student’s participation. Harasim (1991) found five attributes that distinguish this system from other forms of communication. The system allows many-to-many communication. The location of the participants does not matter. The system is asynchronous allowing many people to "talk" and "listen" at the same time. The final attribute is that the system is computer mediated allowing for a neutral view. Because of these attributes, the system works especially well for small group discussions, role playing, debates, peer learning, mutual help, as well as an informal place for socializing, almost like and online café.
With the advancement of online messaging systems such as chat and email, a new word has become commonplace, yet is not practiced by many. The word: Netiquette (Harasim, 1991). Netiquette is a set of unwritten rules of how to be polite online. When communicating electronically, there are rules that should be followed, all equally important. The first is to start with a keyword or subject. Next, keep messages short and limited to one or two screens. In a message, cover only one point, the keyword, per message for clarity. The overall look should also be considered. Double spacing or only upper case can severely hinder the readability of the message. The final point to observe is the tone of the message. Using all capital letters can be seen as screaming while a message with typos conveys a sloppy effort. While in a group, use first names. Reply promptly to questions and add personal comments. However, avoid hostile or offensive text. The use of humor is encouraged, but remember the audience you are "talking" to.
With all the advances of online technology, some classes have begun to be held only in cyberspace. As mentioned above, one school had the quizzes online, yet the students had to take the final exam in person. Online testing is one major area that teachers are looking towards for future use. Some schools have placement exams that help decide what classes students should take. One school has turned to a commercial product to handle their assessment tests (Kayser & Falzone, 1998). At the beginning of the school year, over 800 freshmen needed to be tested and placed. To help with the scoring and paperwork, the computerized system was amazing. Since the computer lab can accommodate 60 students at once, the school set up students in blocks of 60 to take the 45-minute test. The program randomly selected problems from a database and compiled all the scores into the administrative computer system. Even though students were allowed to retake the test, the scores did not change significantly. While the general test used a commercial product, the math department was designing their own test using Visual Basic. The program, entitled HiTest, used a test-template and randomly selected questions from a database. After a question was answered, the next question was based on the result of the previous. As well as recording the student’s questions and answers, each question built it’s own history of correct and wrong answers. Based on this data, "bad" questions could later be reviewed and then fixed or removed.
Kayser and Falzone (1998) also developed a list of advantages and disadvantages when using online testing. Some of the advantages for the student are that they can take the test when they want. They know their results instantly and have an option to retake the test. For teachers, there is more class time and a simplified test creation procedure. Instructors do not have to take time to grade the tests and analysis of the results is automated. Finally, for a class which has more than one section, there is a standardized test. The teacher does not have to become the "bad guy" actually giving the test. Since students can retake tests generated from the database, teachers do not need to create makeup tests. However, creating the test question database can be time consuming. Some of the problems include the fact that the test can only have multiple choice questions. The teacher cannot directly monitor the test being taken. With any technology, "crashes" and problems with networks may always happen. Overall, computerized testing was found to be an effective way of handling placement tests as well as tests to standardize multi-section classes.
In some schools without Internet access, setting up an internal network is a viable option. Electronic bulletin board systems are providing classes with a valuable means of communication (Yanik, Yanik, & Gustafson, 1997). The use of the system varied with each instructor that used the system. The initial setup of the system, a commercially purchased product, was done by a teacher with a fair amount of knowledge about computers, known as a sysop or system operator. The sysop added teachers to the system. Teachers could then add their own class to the system. The teachers mainly used the system for communicating in addition to verbal instruction. Each student had a personal mailbox for mail that was private and could only be read by the student. Another section contained a newsgroup system, or an area to make public postings and receive responses. In addition, the system allowed students to "chat" with other students currently on the system. To encourage use of the system, teacher gave bonus points to students who made postings each week. While reviewing the list of students who posted, teachers discovered those students were the ones who were more outgoing than other students were and asked more questions in class. After using the system for a school year, teachers found that some students used the system in a limited manner while others did not use it at all. One contributing factor was that the system was local, meaning that the students had to be in the lab to use the system. While initial results were somewhat discouraging, the instructors saw potential for use of a similar system in the future.
Overall, there are many ways that teachers can use computing technologies to enhance the learning of mathematics. These technologies can take many different forms from devices small enough to fit in a hand to networks that consist of millions of computers worldwide. The most common application is the calculator. Calculators have a vast range of functions and uses. Simple four function calculators can help students discover how to multiple by looking at patterns of functions, such as 4x1, 4x2, 4x3, and so on.
More advanced graphing calculators can help older students learn and grasp new concepts. The main advantage is that students are not discouraged by having to perform long and complex calculations by hand. Graphing calculators can also help students understand the concept of functions as well as the "visual" aspect of mathematics. Some people even say that math is truly beautiful when graphed properly. The solver function of some calculators, especially, encourages the use of "try and see" logic. Many people feel that this type of constructivism is the best way to learn abstract ideas such as those taught in mathematics. However, sometimes calculators can produce wrong or misleading answers. Also, they can be extremely precise when only a close estimate is needed.
As technology progresses, computers are become commonplace in schools. Within the schools, there are many uses of computers. These computers are used to run programs already created to help students. One of the main advantages is that the computer provides individualized instruction. And with advances in three-dimensional graphics, students’ interests are captured and held throughout the session. Spreadsheets, primary a business application, has also found it’s way into the classroom. Even without using the programs advanced features, the spreadsheet is an excellent visual way for students to organize and display information.
With the increase in the amount of electronic data, traditional floppy disks cannot hold the amount of content needed. Schools have been switching some of their storage and distribution of material to CD-ROM and DVD discs. These forms of media can hold a huge amount of data equaling hundreds or even thousands of traditional floppy disks. One of the large types of files are multimedia presentations. These presentations are presented as a series of slides incorporating text, graphics, and sounds. Teachers must be careful when creating these presentations. The colors, sizes, and fonts can make major differences in the impact and effectiveness of the presentation.
The final computing technology available for teachers’ use is the Internet and the World Wide Web. Some classes are taught completely online with students from all over the world meeting in one place – cyberspace. One of the major problems in working with mathematics on the web is the conflict of how to best display mathematical notations. While some notations are naturally built into the keyboard, the best alternative is the hybrid solution. Plug-ins or small programs that assist a web browser in displaying mathematical notation are the best offer for today’s teachers. With the advancement of online communication, online conferencing or chatting is becoming a very useful technology for teachers. However, this online frontier creates a new form of etiquette and social rules.
The Internet is also a place for helping teachers assess students better. Some schools have used online tests to host assessment tests as well as to equal out multi-section classes. These tests can provide instant grading with students having the option of retaking the tests. Tests are generated randomly from a database of test questions. The questions can also be selected based on the previous answer. As well as creating an equal test, programs can record the past history of the question to help find "bad" or lemon questions. Teachers are always looking for new and better ways to assess a growing student population. Many schools have found that these tests can be very effective and reliable as well as providing valuable data on student performance.
Overall, computing technologies are on the forefront of mathematics education. Computing technologies are allowing students to better understand concepts as well as promoting future exploration without the difficulty of calculations to worry about. Computers are changing the way we communicate in all areas of life. Communicating mathematics electronically is on the cutting edge of technology. Networks, whether local or global, are allowing people to easily communicate and advance further understanding. Current uses of computing technologies are just the precursor to the future. Anyway used, computing technologies are greatly enhancing the mathematics education of students everywhere.
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