Technology & Mathematics
From WikEd
[edit] Descriptions, Definitions, and History
It was not long ago that technology in mathematics classrooms meant using a slide-rule to make difficult computations. In 1972 we were still using tables from textbooks to squares, square roots, and trigonometric functions. Interpolation and extrapolation were valuable skills. With the release of the first pocket sized calculator by Texas Instruments in 1972, things began to change in k-12 math education. Computer use at that time was relegated to university campuses where students were using FORTRAN and COBOL programming languages with keypunch machines and card readers. By 1980 personal computers by Commodore, Tandy, and IBM were making an appearance in many high schools and a few middle schools. Most computer labs at that time were devoted to the instruction of programming in BASIC with very little application to other disciplines. Soon, the development of user friendly software made it possible for math students engage in drill and practice with immediate feedback and math teachers were able to engage the class in simple simulations such as "Lemonade Stand”.
Today, students use hand-held graphing calculators that graph simple and complex relationships from tables and from equations for applications ranging in difficulty from grade school through calculus and differential equations. Computers are used to analyze data, conduct virtual experiments, manipulate virtual manipulatives, and make discoveries through experiences that would have been impossible or at the very least too time consuming to consider feasible in a classroom setting. Technology in the math classroom has evolved from slide rule in 1972 to computer with wireless internet connection, projector, smart-board, and thousands of dollars worth of software. Today, learners have the choice to attend a traditional classroom or the virtual classroom. Toppling market prices in technology, coupled with increasing power potential and popularity, facilitated an invasion of technology into mainstream education (Morrisett, 1996; Westbrook & Kerr, 1996; Means, 1994). For example, in 1981, only 18 percent of U. S. public schools had one computer for instructional use. By 1991, the estimate increased to 98 percent (QED, 1992; Means, 1994; Mageau, 1991). In 1983, the ratio of students to computers in American schools was 125:1. By 1990, the ratio decreased to 20:1 and, by 1995, it reached 9:1. Conversely, other technologies were adopted almost at the same rate. For example, between the 1991-92 school year, only seven percent of schools had CD-ROM drives. That number increased to 37 percent by the 1994-95 school year. In 1991, only one percent of the nation's classrooms had satellite connections, while 17 percent had them three years later in 1994. The number of computers connected to networks in schools climbed from only five percent in 1991 to 28 percent by 1994. About 53 percent of all school districts surveyed reported at least one school connected to the Internet in the 1993-1994 academic year, and 37 states provided network accounts to some 509,000 users in 1995 (Westbrook & Kerr 1996; QED, 1995). Access via computer modem, and telephone connection accelerated rapidly in the past few years, featuring e-mail connections to the Internet. According to a recent survey of Internet host computers (Morrisett, 1996), the survey recorded 6.6 million Internet hosts in 106 countries worldwide. In November 1992, about 279 million messages were transmitted over the Internet and, by November 1994, the number of transported messages reached just over one billion messages, an astounding annual growth of 90 percent. Based on these figures, the world's inhabitants should witness an upsurge to 101 million computer hosts by the year 2000. (5)
As schools continue to acquire more and better hardware and software, the benefit to students increasingly will depend on the skill with which some three million teachers are able to use these new tools. In order to make effective use of educational technology, teachers will have to master a variety of powerful tools, redesign their lesson plans around technology-enhanced resources, and take on a complex new role in the technologically transformed classroom. (2)
[edit] Application in classrooms and similar settings
Mathematics is about 5,000 years old. The first 2,500 years it was an experimental science and belonged to the cultural assets of the Egyptians and other ancient civilizations. Using the above notions, it consisted only of the phases of experimentation and application. About 500 B.C. the Greek took the Egyptian mathematics and applied to it the deductive methods of their philosophy (i.e. they added the phase of exactification), thus establishing mathematics as the deductive science as we know it today. From then on scientific mathematics comprised all three phases. It has become customary to teach mathematics by deductively presenting mathematical knowledge, then asking the students to learn it and use/apply it to solve home work and exam problems. This is as if one would have to learn walking by studying, understanding, then applying scientific descriptions of the muscle motions required for walking – instead of learning by trial and error (i.e. experimentation), as it is done naturally. Most of today’s psychological theories of learning consider learning to be an inductive process in which experimentation plays an important role. Phases of experimentation should complete the traditional teaching methods. This kind of experimentation, performed with paper and pencil, is both time-consuming and error-prone. Within the time available at school, students could produce only a very small number of examples for the purpose of observing and discovering, and a hefty portion of these examples could be faulty due to calculation errors. There is nothing you can observe from only a few, partly wrong examples! From now on algebraic calculators enable students to experiment within almost all topics treated in mathematics teaching. There is no limit to the number of examples the student can do and the electronic assistant guarantees the properness of the results. (1)
The advent of electronic technology has changed the way in which much of mathematics is done today, and this change has allowed teachers and curriculum designers to focus more on mathematical ideas and devote less classroom time to mastery of mechanical and computational skills. Calculators are used both to reduce time spent on routine calculations and to enrich students’ understanding of deep ideas. We know that more sophisticated technology will soon become readily available to the classroom. When symbolic manipulators reach the average classroom, they will no doubt affect how students think and learn about algebraic expressions. As these changes occur, it’s important to maintain sound pedagogical principles, lest the glamour of technology replace opportunities for good learning. In this vein, we will continue to be guided by the fundamental principles that the student’s use of technology in the mathematics curriculum should be grounded in concrete experience and that it should serve a meaningful problem–solving purpose. (3)
According to Harold Wenglinsky, "One of the real benefits of different types of technology is the way they influence how teachers and students relate to each other." Technology also can liberate teachers from the traditional lecture-style of instruction, by encouraging them to act instead as coaches and facilitators. (4)
[edit] Evidence of effectiveness
Although our understanding of the effectiveness of various applications of educational technology remains incomplete, such research as is available, combined with anecdotal reports of the positive experiences of a number of schools, suggests that technology may indeed have the potential to play a major role in transforming elementary and secondary education in the United States. While a critical discussion of the existing research literature (and of the need for additional research) will be deferred until Section 8, a few of the better-known examples of the successful application of technology to K-12 education may help to convey an intuitive feeling for the potential of educational technology:
Blackstock Junior High School (California): This school has ten "smart classrooms," including one in which students can use computer-aided design (CAD) software to describe products that are then fabricated using a computer-controlled flexible manufacturing system. Higher test scores and improvements in comprehension, motivation, and attitude have been reported for the predominantly Hispanic student body.
Carrollton City School District (Georgia): Computer technology is used in this school district as part of a novel program that has succeeded in reducing the dropout rate from 19 percent to 5 percent, and the failure rate in ninth grade algebra from 38 percent to 3 percent.
Carter Lawrence School (Tennessee): Students in selected classrooms within this Nashville middle school used technology in various ways as part of a program called Schools for Thought, which is based largely on constructivist principles. Sixth-grade SFT participants scored higher on a number of components of Tennessee's mandated standardized achievement test than students in matched comparison classrooms, and demonstrated substantially stronger critical thinking skills in complex performance assessments involving high-level reading and writing tasks. Absenteeism and student withdrawal rates were also dramatically lower among SFT students.
Christopher Columbus Middle School (New Jersey): Perhaps the most widely publicized example of the successful application of educational technology, this inner-city school in Union City implemented a reform program that (along with other important changes) provided all seventh-grade students and teachers with access to computers and the Internet, both at school and at home. The performance of its 91 percent Hispanic student population, the majority economically disadvantaged, improved from significantly below to somewhat above the statewide average in reading, language arts, and math.
Clearview Elementary School (California): A restructuring program involving the use of advanced technology resulted in an increase in standardized achievement test scores from the lowest 10 percent to the highest 20 percent.
East Bakersfield High School (California): A school-to-work program at this school has made extensive use of technology to provide its 60 percent Hispanic student body (including many students having very limited English proficiency) with the skills required for any of five different career tracks, resulting in increased graduation and job placement rates.
Northbrook Middle School (Texas): Interdisciplinary teams use computing and networking resources to teach critical thinking and problem-solving skills to this student population, which consists primarily of the children of migrant workers, 76 percent of whom are economically disadvantaged. Highly significant increases in test scores have been reported.
Ralph Bunche School (New York): Information technology has been used for collaborative work and project-oriented learning by 120 randomly-selected students in this elementary school, which serves primarily low-income black and Hispanic residents of Central Harlem. These students outperformed a control group by ten percentage points in mathematics on New York City standardized exams. Progress has also been reported on problem-solving skills.
Taylorsville Elementary School (Indiana): Self-paced individualized learning is the central focus of this suburban school, whose students are drawn largely from lower middle-class white families. Technology is used to support project work conducted by teams that include students of a mixture of different ages. Internet access and sophisticated information retrieval tools are used to support self-directed inquiries. While the program is relatively young, some improvement has been reported in test scores, along with a significant increase in student interest and enthusiasm for learning.
Rigorous, systematic, well-controlled research will ultimately be required to identify the specific factors responsible for such apparently successful outcomes and to ascertain their range of applicability and the extent to which they can be generalized. Most researchers and practitioners in the field of educational technology, however, are already convinced that information technologies have the potential not only to improve the efficacy of our current teaching methods, but perhaps more importantly, to support fundamental changes in those methods that could have important implications for the next generation of Americans. (2)
Harold Wenglinsky’s study, "Does It Compute? The Relationship Between Educational Technology and Student Achievement in Mathematics," is based on the performance data of 4th and 8th graders who took the math section of the 1996 National Assessment of Educational Progress. The study will also be published as an ETS policy information report, and is available here.
Administered by the U.S. Department of Education, NAEP--known as "the nation's report card"--has tracked student achievement for nearly three decades. For the first time in 1996, NAEP asked students and teachers additional questions about how they used computers in math.
After factoring out the influence of several other variables that affect achievement, such as students' socioeconomic status, class size, and teacher qualifications, Wenglinsky found strong links between certain kinds of technology use, higher scores on NAEP, and an improved school climate. In every case, the gains were greater at the middle school level than in elementary school.
Among his findings: Eighth graders whose teachers used computers mostly for "simulations and applications"--generally associated with higher-order thinking--performed better on NAEP than students whose teachers did not., and 4th graders whose teachers used instructional computers mostly for math/learning games posted an achievement gain equal to roughly 15 percent of a grade level. (4)
[edit] Critics and their rationale
In the same study as noted above, Harold Wenglinsky’s study, "Does It Compute? The Relationship Between Educational Technology and Student Achievement in Mathematics," 8th graders whose teachers used computers primarily for "drill and practice"--generally associated with lower-order thinking--performed worse, and the research found no association, positive or negative, between 4th graders' scores and either simulations and applications or drill-and-practice.
Students who spent more time on computers in school didn't score any higher than their peers; in fact, they performed slightly worse.
The same factors that were tied to better achievement also appeared to be linked to an improved school climate. Where teachers had professional development with computers and used them for teaching higher-order skills, schools tended to enjoy higher staff morale and lower absenteeism rates.
Critics and some skeptics have long argued that computers add nothing to the education process, and may even be a distraction. (4)
Unfortunately, most of the nation's schools aren't using computers in ways that Wenglinsky's findings indicate are linked to better scores. And survey responses from the 1996 NAEP also raise serious questions about technology's role in closing the achievement gap between disadvantaged students and their peers.
Over the past decade, tremendous resources have been committed to the cause of ensuring that schools in disadvantaged communities have computer equipment and network connections on par with those serving more affluent populations. Survey results show, for example, that black students use computers in learning math somewhat more often than white students.
But the survey paints a very different picture about how those students use their schools' computers. At the 8th grade level, about 31 percent of white students used computers mostly for simulations and applications, compared with just 14 percent of black students. At the same time, more than half of America's black students had teachers who used computers mostly for drill-and-practice, compared with only 30 percent of white students.
In short, black students have closed the digital divide where it matters least--the amount of time on a computer. The gap persists where it matters most--how the computer is used. "It's the low-income communities that have invested the most in technology for drill-and-practice," says Margaret Honey, the deputy director of the New York City-based Center for Children and Technology. "But if this research says those investments don't have the kind of impact we want them to have, then that's an important message. "There's an expectation among many urban administrators that kids from disadvantaged backgrounds need basic skills and nothing else," she adds. (4)
[edit] Alternative explanations due to Diversity or Socio Economic Status considerations
Specifically targeted federal programs have in recent years helped to substantially mitigate some of the disparities in access to educational technology that had earlier been associated with socioeconomic variables. Income-related differences in computer density, for example, have been reduced to a relatively modest (though still not insignificant) level: During the 1994-95 school year, the poorest schools had one computer for every 11 students, while each computer in the richest schools was shared by 9.5 students.
While this progress is certainly encouraging, there are several reasons for continued concern.
Factors still exist that may be contributing to the disadvantages experienced by low-SES students.
The substantially lower prevalence of computers within the homes of low-SES students may be among the most difficult forms of inequity to remedy. As of June 1995, computers were present in only 14 percent of all households headed by adults who had completed no more than a high-school education, and in which annual household income was less than $30,000; the comparable figure for households headed by college-educated adults having a combined income of more than $50,000 per year was more than five times greater, at 73 percent.
Wealthy school districts may be able to recruit teachers with greater expertise in the use of educational technologies by offering above-average salaries, or to offer their existing teachers more technology-related training and technical support. Poorer schools, on the other hand, may have fewer teachers capable of making effective use of educational technologies, thus limiting both the quality and quantity of computer use by their students.
As interactive information technologies come to be used increasingly for school work and other forms of learning, SES-linked differences in the ownership of home computer systems threatens not only to perpetuate existing familial patterns of socioeconomic disadvantage, but to widen the gap between the most and least affluent Americans. At a time when U.S. income inequality has reached its highest level since 1947 (when the Census Bureau began monitoring the relevant index), the educational implications of SES-related disparities in home computer ownership should be regarded as a source of serious concern from a public policy viewpoint.
[edit] Signed "life experiences", testimonies and stories
Keep in mind that the goals are to learn the mathematical concepts not to learn how to use the technology.
I teach two classes that include technology and mathematics. One class students learn the calculation and design of heating, ventilation and air conditioning(HVAC) for commercial buildings. The other class is sheet metal fabrication. Both these classes help students design, layout and fabricate sheet metal for HVAC equipment. As instructors, we spend a fair amount of time, more than we would like, on Algebra and Geometry equations related to these classes. I found this article very insightful as to the reason why students have difficulty grasping the application for the equations we use. Electronic technology has changed the way in which mathematics is taught today. I look forward to exploring different and newer avenues for teaching with technology. -Kent Randall
After reading your info, it brought back some great images of the Apple IIe computers we had at our school. Computers then were used as playing games to learn. We use to think that the students were learning so much while playing those games like lemonade stand. I don’t know if I ever imagined the impact computers would have in our society. Now they have a more business type of learning involved. I think it is essential that each student learn the functions of program like MS Office and all the capabilities it has. Students need to learn that a spreadsheet can be used for both math functions and for organizing data. Computers should play a huge role in educating our youth, not just to play educational games, but also to use to develop life long skills which we as adults will depend on. – Dale Donner
Hi. My name is Melanie Smith. A few years ago I took a class called "Technology and Mathematics". The course was designed to show us some of the different technology out there to help us teachers develop lessons, demonstrate material to our students, and help us connect more with todays generation. I found the class very enjoyable and insightful. It wasn't about how we should be using technology to teach our students, but how we can use technology to aid us in teaching our students. We learned about different software to help teachers create worksheets, lesson plans, and exams, such as Microsoft WORD, Geometers Sketchpad, MAPLE, and Mathematica. I found the class very useful and to this day I use some of the software that we learned about in that class!--Melanie Smith
I have used Geometer's Sketchpad in my classroom and it is a wonderful program. Although, most of what I used Sketchpad for could have been taught without the use of Sketchpad, the real benefit to the program was the way the students reacted to it. Half the battle in teaching math is convincing students that they can do it. With Sketchpad, students are not only able to complete the sketches but they are able to discover geometric properties for themselves. Students tend to remember concepts better when they discover them. Also, students don't get bored with Sketchpad like they do sitting in a classroom with a textbook. If I can keep their interest, I am more likely to be able to teach them something. Another advantage to Sketchpad is that students must understand the correct mathematical terminology in order to create the sketches. It is great to reinforce the terms without student's realizing that every move they make in Sketchpad has a hidden vocabulary test incorporated! I love it when student's learn when they think they are playing! -- Rita Grunloh
My dual certifications in history and math, has allowed me to connect subject areas and teaching methods that would ordinarily remain separate. In fact, technology has allowed me to incorporate mathematics into my social studies curriculum. For example, before drafting their own national policy to combat the spread of HIV my high school seniors utilized modeling software called Netlogo to study how the HIV virus spreads. Likewise, my students have tracked the cost of their personal use of natural resources using excel while learning about conservation of renewable/nonrenewable resources. Although the potential for technology may appear limitless, its effectiveness is never guaranteed. Technology can be the greatest aid or hindrance depending on how it is implemented in the classroom. All technology should be used only after careful examination. While computer models, imaging programs, and calculators can aid in making the impossible a reality they can also lead to student reliance. As teachers we have the responsibility to remain updated on educational applications of technology and their effective implementation. -- Brian Zeglin
The graphing calculator is a required tool for the mathematics courses I teach, probability and statistics and AP statistics. However the students that come to these classes have had little practice with the calculator. As the calculator becomes a more powerful and common tool we need to make sure that we integrate it into lesson planning to ensure that our students are using the calculator to the fullest extent. -– Matthew McCulley
While using calculators (or other forms of technology) can speed up a mathematical process, we must be certain that our students understand what we are teaching them. In my senior-level statistics course we start by doing the problems by hand, which tends to be lengthy and time-consuming. The next day, or later in the same period, I introduce them to the aspects of the problem that can be completed on a calculator. In this way, the students are learning the basic concepts, but are also learning that a calculator is something that is an asset, not a requirement. M Foshee
I am a first year teacher in Mathematics and find it essential to use technology in the classroom especially with the growth of technology in our culture. Using graphing calculators helps students visualize some concepts that might be hard to picture and can speed up the mental math process to allow for more instructional time. Technology is growing so fast and we can use new instructional tools. My personal favorite is the wireless tablet to use with a computer and a projector. This allows the instructor to not be glued to the front of the classrom and can teach from any spot. This helps aid in classroom management as well. Another neat tool is TurningPoint clickers. These are useful to assess students in a formal or informal way on thier understanding of current material. Instead of orally responding to questions, each student has a clicker and can "click" their answer in. The computer calculates thier responses and shows the results right there on the screen. It enhances the learning and takes students out of the ordinary pencil and paper assessment and verbally responding to questions. This also allows for students to remain annonymous. Sometimes students are hesitant to participate in the class because they are afraid to get the wrong answer or that students will make fun of them. There is so much technology out there and I encourage teachers to learn and explore the various tools. E. Kaffel
[edit] References and other links of interest
1. The Algebraic Calculator as a Pedagogical Tool for Teaching Mathematics by Bernhard Kutzler: http://b.kutzler.com/article/art_paed/ped-tool.html
2. Report to the President on the Use of Technology to Strengthen K-12 Education in the United States March 1997 PRESIDENT'S COMMITTEE OF ADVISORS ON SCIENCE AND TECHNOLOGY http://www.ostp.gov/PCAST/k-12ed.html
3. Lynne Alper, Dan Fendel, Sherry Fraser, Diane Resek, Technology in the IMP classroom http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Resek/Resek-1.html
4. Jeff Archer, The Link to Higher Scores, Education Week on the Web, 1998 http://counts.edweek.org/sreports/tc98/ets/ets-n.htm
5. Mohammad Khalid Hamza and Bassem Alhalabi, Technology and Education: Between Chaos and Order, First Monday, Peer-Reviewed Journal on the Internet http://firstmonday.org/issues/issue4_3/hamza/ [*http://schoolcio.com/schoolcio/whitepapers/FranklinRegional2006.pdf Technology and Learning Magazine Using Whiteboards teaching Math]
