John A. Goebel
Daniel J. Teague
NC School of Science and Mathematics
Incorporating Technology into the Curriculum
at the North Carolina School of Science and Mathematics
The Mathematics Department oft the North Carolina School of Science and Mathematics has made widespread use of technology in the classroom for the past fourteen years. While we have somewhat of a unique situation, we feel that most of what we have done with technology can be done in a normal school setting. In this time period, we have learned some lessons about the use of technology in the classroom, but before we share those lessons, we need to give some background into how we have gotten to where we are.
Where we are and how we got here.
The North Carolina School of Science and Mathematics (NCSSM) was established with two primary goals. The first was to give the brightest students in North Carolina an opportunity to study science and mathematics courses that they might not have the chance to take at their smaller schools scattered across the state. The second goal was to create an environment where science and mathematics teachers could develop new curriculum and effective teaching strategies and share these with teachers around the state and nation. When NCSSM opened in 1980, technology did not play a major role in the program. Computers were around, but little software existed that was useful for the teaching, learning, or doing mathematics. Graphing calculators did not exist. The Internet existed, but without a graphical interface. The World Wide Web did not exist. The curriculum we offered our students in the early 1980's was founded in the Agenda for Action, with an emphasis on problem solving and mathematical modeling, but little or no use of technology.
In 1985 the Mathematics Department received its first major grant, this one from the Carnegie Corporation of New York, to develop a precalculus curriculum that took advantage of the technology that was beginning to emerge. At this time that meant the scientific calculator and the computer. As we worked on this project, we found that affordable software did not exist that could graph functions, fit curves to data points, and do matrix arithmetic. We wrote software to perform these tasks. Before we finished our first curriculum project, graphing calculators had hit the scene.
Before looking at the impact that technology has had on our program since these early days, we must set the scene for these changes. Our school has near ideal conditions for implementing new curriculum ideas and new technology. First, we have exceptional students (average SAT score about 1350) who are in residence at the school. Our students do not have jobs or cars to distract them, although they are active in clubs and athletics. Second, we have a collection of talented mathematics teachers who are dedicated to both of the missions of the school, do give our students the best courses we can and to stay at the forefront in curriculum reform. We have received several grants to develop programs, but independent of these projects, we make the time to meet with each other to discuss mathematics and methods of teaching. If any one of us has a good idea or has a problem with content or methodology, we have colleagues to turn to for help. Next we have been able to require our students to have their own graphing calculator. We have better computing facilities that most schools, but like most schools, we find it a constant struggle to keep both equipment and software up to date. Finally, we have many contacts, both with organizations like NCTM who are constantly searching for the best way to education students and with businesses who are producing technology like calculators and software for use in the classroom. With all of these things in our favor, we most certainly have advantages that most schools do not. We are aware of these advantages and believe that we have the responsibility to experiment, to develop programs, and to share what we learn with any and everyone who will listen. We are not making these points to brag about our school, but rather to point out that we have an ideal situation in which to implement change, especially change involving technology. We are certainly not the typical public high school when it comes to our mathematics faculty or our facilities. If we cannot make changes and incorporate new ideas and technology into our program, others with fewer advantages will find it much harder.
Our first curriculum project, to develop a contemporary pre-calculus course, was based on the premise that technology would soon be available that would substantially change both the content of the traditional precalculus course as well as how students learn and do mathematics. Of primary importance was the ability to graph functions. The ability to have a reliable graph of a function has meant that we could concentrate on creating functions that model phenomenon and not spend so much time worrying about how generic functions behave. Modeling applications has become the emphasis of this course. A part of this modeling process also involves gathering data and being able to fit a function to that data. Our precalculus course does not have data analysis as an add-on; it is an integral part of the modeling process. Finally, the ability to find the inverse and power of a matrix has enabled us to actually use matrices in realistic problems, such as Markov chains and Leslie population models
We need to make it clear that we were not developing this precalculus course just to teach at the North Carolina School of Science and Mathematics, indeed, not even a course just to teach to the very best students at any school. We believed then and still believe, that every student going to college needs a course in mathematics that will prepare him or her for a broad array of college courses. The first name for our course, "Introduction to College Mathematics", was an effort to convey that this course was for any student who planned to go on to college. We anticipated that technology would play an important role in this course over the coming years, but we were not sure how extensive a role this would be. Initially all we had was a computer with a large monitor in class for class demonstrations and explorations. This was the extent of our technology. We hoped to be able to explore functions, data analysis and matrices in class with the students. At a minimum, we hoped that other classes that used our material would have a computer in the classroom to explore ideas graphically and numerically in class. We were fortunate enough to get additional computers and set up several computer labs where our students could explore ideas on their own outside of class. To do this, our students used software that we wrote as affordable and user friendly software did not exist. After our material went to the publisher, but before it appeared in print, graphing calculators began to appear that would do much of what we wanted to do. We wrote calculator programs to do those things that the calculators still would not easily do, such as finding residuals for regression fits. In those early days and continuing to the present, we have always wanted our students to have access to any technology, hard or soft, that we used in class. The graphing calculator addressed this for the most part.
As we are preparing the second edition of our precalculus materials, we find that technology has moved faster than we have been able to react and revise. Our first edition had very little about recursion. As we explored topics in our Fractals class and topics like like Euler's and Newton's methods in our Calculus classes, we felt that recursion was a topic that we needed to include in our revision of Precalculus. Computers have always been able to do recursion well, but the fact that the current versions of most calculators also do recursion very easily, meant that we could include as much recursion as we saw fit. The temptation, when revising curriculum with technology in mind, is to include topics that the technologies can do well, often without regard to their relative importance in the curriculum. We feel that the topics we have included in our precalculus course are important and have always been. The technology is now enabling us to teach those topics more effectively. The idea of change has always been important. Before technology, the study of change was left almost entirely to the calculus and differential equations courses, and even there it was easier to use closed forms and continuous functions to study change. Students were often given the functions and asked to describe the change, or given the change and forced to learn difficult methods to derive functions from rates of change. With the ease of computing values recursively, we can now explore change at a much earlier point in the curriculum in a way that more students can understand and remember.
Data analysis is another topic that has always been important to mathematics and science. We are not teaching this now just because the technology is there to do it. It has always been important to use a function to model a phenomenon. Until recently, however, it was beyond the capability of the average student to gather data and to analyze the data to find functions as models. Instead we were given the models and taught and learned how to manipulate the model to make predictions. Now, with the aid of technology, we are able to gather our own data and efficiently fit a curve to that data. The process of finding the model helps students understand the phenomenon and the predicting steps are more meaningful. The mathematics needed to find the model in more important than the mathematics required to manipulate final form of the model.
Our second project, a calculus reform project that we undertook in cooperation with Duke University on 1989, drew on the strengths of our precalculus course and extended the use of technology in a more formal way. For this course we wrote labs and investigations into the course that required extensive use of the computer. As graphing calculators have improved, many of these labs and investigations can now be done effectively using this technology.
We felt that it was entirely appropriate to plan for the use of technology, mainly computers, in a more formal and extensive way as we developed our calculus materials. First, in all schools fewer students take calculus, so it would be reasonable that smaller computer labs could handle calculus classes. Secondly, technology was advancing at a rapid pace, and even hand-held calculators were capable of symbolic algebra. The technology, whether it was a computer or calculator, enabled students to explore concepts in both graphical and numerical ways - ways that are easier for many to grasp but much harder to implement without technology. The labs and investigations that we wrote into our calculus course fall into two categories. Many of the early labs are exploratory, with students looking at a concept in different ways and drawing conclusions. The derivative, both the concept and its uses, limits, and local linearity, are explored in this way. The later labs and investigations require students to use the calculus they have learned to solve realistic and sometimes involved problems. These are typically problems that require graphical, numerical, and analytical solutions and involve more than one simple concept.
Our other courses, including Fractals and Chaos, Discrete and Finite Mathematics, Explorations in Geometry, Multivariable Calculus, and Statistics all incorporate the use of technology, some more than others. Each of these courses is, however, very different than they would be without the use of technology.
More recently we have taught mathematics classes via distance learning. These courses, which range from year-long courses taught entirely by one of our teachers, to team-teaching collaborations with other teachers, to short workshops for students or teachers, are made possible by the two-way audio and video connections that are a part of the North Carolina Information Highway. We have offered Precalculus, Calculus, Multivariable Calculus, Statistics, and Algebra to student across North Carolina. Students that the remote sites all use graphing calculators, some use computers and software to do mathematics and to communicate with our teachers. In this area we have few contemporaries to learn from. To date the classes we have developed for our residential program have determined what we have taught via distance learning. There has been minimal impact on our residential program as a result of distance learning technologies, but this may be about to change. As a part of distance learning, we are beginning to use Web pages to disseminate information to our distance learning sites. Students at these remote sites can access daily lessons, download labs and programs, and submit their solutions via the Internet. Our own students, that is, those in our on-campus classes, are also finding these sites and using them, even without our direction. We think this medium has the potential not to replace what we do in the classroom, but to supplement our classroom activities. For this discussion, I am placing CD-ROM interactive lessons in the same category as Web Pages, as Web pages are mimicking early hypertext CD's and vice versa.
While it is too early to know the impact of Web Sites and CD-ROM courses, we do have some thoughts about their use. First, these media are simply additional forms for presenting information. Certainly with animations and multi-media, they may be more appealing than traditional textbook courses, but the glitter will wear off. We have not read anything that would indicate that schools or universities feared that the printing press would put them out of business. If such fear existed, it was unfounded. We think there will be an initial bubble in which some students will buy these CD's or access course-based web sites and think that they "know" geometry, for example, because they zipped through a geometry course on the computer. Except in the rarest case, students will still need guidance, structure, and some external evaluation to ensure that they know what we as professional mathematicians think they should know. We have already seen this phenomenon with some of our students and students we have worked with on our State Math Team. Many are very bright and have "taught" themselves various courses. Because the are working in isolation, they are generally unable to determine which ideas are really important and which are secondary. In addition, they form misconceptions and have no one to correct them. Working with other students and a teacher, there is a greater likelihood that topics are put into some perspective and misconceptions are minimized.
I think that Web-based courses that have a person to communicate with and some form of external evaluation procedures are likely to be more useful than stand-alone CD's. The hybrids, those courses which combine CD's and Web sites, would seem to be more useful, but only if the student taking the course can interact with a person, give input, get feedback, submit work, and be evaluated externally.
Lessons We Have Learnt and Recommendations
Technology Changes Too Rapidly to Keep Up:
Technology does not upgrade itself. Technology is changing very rapidly and it is nearly impossible for schools to upgrade equipment and software fast enough. Even if schools could afford to keep hard and software up-to-date, it would be nearly impossible for teachers to learn the technology and think about the best way to incorporate the latest technology into the curriculum. Often the next calculator or next version of software is out before we as teachers have learned the previous version. Often students have better technology than we do and can't understand why we are behind. We lose credibility.
Recommendation: The Standards should set realistic goals for implementing technology in the classroom. Along with this school districts need to know that equipping the mathematics classroom is not a matter of throwing a box of chalk to the teacher. Mathematics classrooms must have calculators, overhead projection systems, and computers for students to use in the classroom.
Technology doesn't teach itself. We cannot assume that every student (or teacher) will be able to take a calculator from the box and start using it effectively and efficiently. It takes time, often time away from the mathematics content, to learn how to use the technology. This is even more true of the software that is available to do mathematics. Often, but not always, the time spent learning the technology is made up later in the course, since use of technology can save time on other problems. The time required to teach students to use the technology must be a part of the syllabus, particularly during the beginning weeks of school. This may give the impression of "losing ground" if movement through the course is measured against years without technology. You may not have covered as much content at the end of the first quarter or semester.
Recommendation: The Standards must stress that teacher pre-service and in-service training includes the appropriate use of technology. As teachers usually teach the way they were taught, technology must be used in the mathematics courses that prospective teachers take.
Recommendation: Technology must be taught along with the mathematics in the classroom. We cannot assume that students will just pick up the appropriate use of technology without our example and our teaching.
Notation is often difficult for students. This becomes a bigger factor with technology, since the technology is unforgiving. It is often the case that the notation needed for the technology is not the same as that used in the texts. The teacher must either alter the notation used in class, or deal with both notations. In our texts, we would prefer to write iterative equations as , describing future values in terms of present values. This seems to make more sense to students. However, we used the poorer notation , describing present values in terms of past values because that is the notation for the calculators used in the course. One side effect of using technology might be more careful attention to notation in our teaching and routine work with students. Many of us are far too sloppy with notation and far too willing to give "partial credit" on paper for anything that looks close. The computer will not accept "close."
A second issue of notation is the difference in two-dimensional mathematical notation on the paper and in the text, for example, the linear notation required for calculators and some computer software, . This linear notation is a significant problem when first using technology, and students who are unfamiliar with both forms of the notation will have a difficult time learning the mathematics. More of the software and even the hand-held calculators are moving in the direction of making their math look like the math we write on the paper, so this may become less of an issue.
Recommendation: Proper notation is more important than ever in mathematics. Software developers should implement standard notation, and students should be taught to use proper notation.
The Big Picture:
Technology allows students to focus on the "Big Picture", but also allows them to lose this focus, particularly when they are learning to use the technology. Teachers need to realize that the student's focus will be to "get the technology to perform a certain task", focusing all of their energy on that goal. In the process, they will lose sight of the reason they wanted the technology to perform the task to begin with; they lose sight of the mathematics. Being aware of this, the teacher must constantly remind them of the bigger picture, in which this particular task is only a small part.
Students must understand the problem to be able to give order to the technological steps used in generating the solutions. Understanding of the big picture is not enough. They must understand how all the small pieces of the problem fit together. Often the difficulties with technology are in the details.
An additional difficulty is when solving an equation using technology, often only the principle solution is found. Students who are focused on the technology and not the problem will stop with this first solution, even if it is not in the domain of the problem, rather than continue to find other or more appropriate solutions as well. This has always been an issue, even when using tables to find values, but students' trust in the computer causes them to believe whatever answer the computer gives them, even if it is completely unreasonable.
Recommendation: Technology should be incorporated into the daily teaching, learning and doing of mathematics so that students learn when to and when not to rely on the technology and when to believe answers that the technology gives us. As we are able to use more complex and interesting problems in the classroom, more attention needs to be paid to the "big picture." Students need to learn that "getting an answer" is not the sole purpose of problem solving.
Students need to develop the habit of documenting the work the do with the help of technology. They lose equations when they look at the graph, then lose the graph when they look at the equation. Being able to document intermediate steps allows them to repeat the process later. The requirement that they be able to document their work makes them better able to explain what they are doing and why they have chosen to do it. Some mathematics software allows students to include text, graphs, and mathematical expressions in the same document. This gives the students the opportunity to annotate or document each step in the process of problem solving. Few do, and those only under much duress.
Recommendation: As the mathematics problems we do become more realistic and less algorithmic, students must be required to write explanations to their solutions. Solving a problem and making a valid and convincing argument should be our goal.
The importance of "real-world problems" is enhanced with technology. Since there is less of a need for problems to work out "nicely", the students expect that the problems will be realistic. They are disappointed in made-up data, for example. This requires more time for the teacher in preparation. Technologies that enable the student to gather their own data reinforce the desire to gather real data, but data-gathering activities are very time consuming, taking away from the traditional instructional time. We are not all in agreement that student gathered data is any better than real data found in textbooks or on the Internet.
Students don't automatically make good choices about what technology to use (calculator or computer, for example) and when to use technology and when to use paper and pencil. This choice-making must become part of the curriculum. The dependence on calculators seems to enhance students' lack of number sense and estimation skills. As mentioned earlier, students trust technology so they are too willing to accept unreasonable answers.
While technology opens the door to many new and exciting possibilities, it also makes teaching much more difficult. There is less certainty involved in teaching now than ever before. Teachers have far more choices of what to teach and how to teach it, and not everyone agrees. Technology is changing so rapidly that, even if you want to include the latest technology in you classroom, it is difficult to keep up. When we as teachers at a special school like NCSSM feel like this, we know that for many teachers the task is much more difficult.