Robert L. Devaney
Department of Mathematics
Boston, MA 02215
Technology influences many areas of mathematics. One particularly important topic that is affected in many ways and at many different grade levels is solving equations. In the past, the only tool that users of mathematics had available to solve equations was algebra. However, when the equations involved are nonlinear, algebra usually fails miserably to solve a typical equation. Now, however, technology comes to the rescue with readily available geometric, symbolic, and numerical methods to "find" solutions. We discuss a few of these methods in this paper, discuss how the curriculum may change in order to accomodate them, and give an example of a course based on solving equations that radically changed in the past ten years.
In the ancient times (ten or fifteen years ago), when a scientist or engineer had to solve an equation, there were simply very few tools available. He or she could hope that the equation was "nice" and had some easy-to-guess solutions. Or perhaps the equation featured some obvious symmetry that would reduce it to a simpler solvable equation. Or maybe the equations fit into some very special format that some ancient mathematician had dreamed up a clever solution method for. But that was about it; there simply were not many tools available for solving equations in those dark and dreary days.
I'm not necessarily talking about solving simple polynomial equations here. All sorts of scientific problems reduce to "solving equations." The equations can be algebraic equations or systems of nonlinear equations, differential equations or partial differential equations or integral equations: but the question is always the same. Find a solution. Or find all solutions. The solutions may be numbers or vectors or functions, but the goal is to find that missing piece of information using some sort of formula.
Now technology changes all of this. Instead of hunting for a formula or function that "works," the scientist or engineer has many different options available. Our goal in this paper is to describe some of these options, to show how some fit into the curriculum, and then to describe how one such "equation solving course" at the college level has been profoundly impacted by technology.
1. What we do now to solve equations. If a working scientist or engineer encounters an equation af any type today, the first thing that he or she will attempt to do is turn to an "equation solver." This may be Mathematica or Maple or some other computer algebra system that can solve difficult equations much faster than humans can. If there is a formula for the solution, it's a good bet that the machine will find it. That's not to say that the computer will find the "best" solution, or the "most important" solution (whatever that means). But it is likely that the computer will do the job much more quickly and efficiently than the human will. At this point, these computer algebra systems are still a little too complicated for use in secondary schools. But things are changing fast, and I suspect that it will not be too much longer before every student will have access to a powerful and easy-to-use solver.
2. So why should we bother to teach equation solving? Obviously, we need to teach students how to solve some equations just so they know what solutions mean. They should not run to the computer to solve linear equations such as x = 3x. Or the equation x^2 =3x. Or dx/dt = 3x. And everyone will argue about their favorite type of equation that students absolutely MUST know how to solve by hand. Fine. Teach them. Maybe even come to a consensus on what these equations are. But do we need to spend endless hours teaching specialized tricks to solve equations? Do we need to work for hours to reduce a particular rational expression to one that, in the old days, was considered nice and neat? My answer is an emphatic no. There are much better things that we can now teach instead of these manipulations.
3. What should replace the manipulative skills? Just as is being championed in the reform calculus movement, qualitative and numerical issues regarding equation solving are now much more important. In terms of qualitative behavior, we can now view the graphs of polynomial equations with ease and we can estimate where the roots are and figure out how many there are. We can draw the surfaces generated by nonlinear systems of equations and see how and where they intersect. We can view the slope fields for differential equations and use this picture to get a hold on how solutions behave. All of this gives us a very different tool for understanding what solutions are and where they are.
Then, if we really need the exact numerical answer, we can turn to numerical methods to find solutions explicitly.
4. Numerics in the curriculum. This is another topic that has been woefully neglected in the traditional mathematics curriculum, simply because we never had the wherewithal to make the thousands and millions of computations necessary to utilize typical numerical method. I would argue that now is an ideal time to bring back numerical algorithms to the curriculum. Remember the old Babylonian method for finding square roots? That seemed to disappear as soon as scientific calculators made their appearance. But I believe that this is a natural method to incorporate into the curriculum to give students an idea how machines actually do the computations. In algebra classes we can teach students the iterative algorithm for finding the square root of K---recall that the algorithm says to make an initial guess, compute the average of your guess and K/guess, and then iterate. The fact that this can be put into a spreadsheet or into an iterative calculator frees students from concentrating on the mindless manipulations and let's them see firsthand how quickly this algorithm always finds the root.
Now many people would argue that we got rid of this for an appropriate reason, that technology finds square roots for us now. Yes, that's true, but read on....
This method can be extended to finding nth roots in any of a number of ways. Students can experiment with different algorithms and determine which converge faster (or at all). Put in a guess and watch the spreadsheet compute the results. Change your guess and see that the same result occurs. Now purists will say that using the spreadsheet will further erode the students algebraic skills. But I maintain that just entering the formulas into a spreadsheet correctly is a valuable mathematical enterprise, and if students can do this effectively, they really have "gotten it." More to the point, they are learning how a practicing scientist or engineer works with a problem. There may not be a button on your calculator for fifth roots, but there is an easy mathematical recipe to find these numbers. Employing the recipe necessitates that the user
This to me is exactly what a user of mathematics will need to know in the future, and here we can teach it at relatively low levels.
Other numerical methods or recipes abound throughout the curriculum: simplex methods, Newton's method, numerical methods for differential equations, recipes for solving the Towers of Hanoi and related puzzles, and so forth. All of these can and should be taught with the computer as a tool, to help with the visualization and to provide the computational power for the numerics.
5. How my course has changed. Let me describe the incredible changes that have occurred in a course that I often teach. This is a differential equations course at the college level. Now I know that the content and student population of this course has nothing to do with the issue of Standards 2000. However, this cours IS an equation solving course. And I anticipate that our experiences in this course will be shared by many algebra teachers down the road.
The differential equations course at most colleges and universitites is really calculus 4. Virtually every engineering, physics, and mathematics major takes this course. Now that is not a huge population as in the basic calculus or algebra seqeunce in secondary school or college, but it does represent a large "service" course for most universities.
In those ancient days, this course was the closest thing to a cookbook course in the mathematics curriculum. My class used to consist of 36 class meetings, and, curiously, the course consisted of 36 tricks to solve differential equations. Some were important: methods to solve linear equations for example. But some were so obscure that I could not remember how to apply them by the time I finished teaching the course. And I had taught the course six or seven times! It is no wonder that the students retained practically nothing from the course.
Furthermore, the students who did well in the course were the kinds of students who could compute very well. These students possessed excellent algebraic skills; they could differentiate 50 functions in less than five minutes; they could integrate any function I threw at them; but when they finished that old course, they had little idea of what a differential equation really was. And they had no real idea of what a solution did; yes they got the formula right, and all the +C's were in the exact right location, but if I ever asked tehm what this corresponded to in the real world, they would have no idea whatsoever.
In our new course, most of these methods of analytic solution are gone. For those who remember the course, topics like power series solutions, Laplace transforms, variation of parameters, exact equations, and (fortunately) I forget the rest of the tricks, are now gone. In their place are more qualitative and analytic methods. The computer (actually two of them) isalways running during the class sessions. I never put a differential equation on the board without using technology to view the slope field or phase plane, and I never finish an equation without asking the students to explain what the solutions mean for the original physical problem. Does the spring oscillate as it winds down? Does the population explode or go extinct? Does the voltage in across the capacitor reach equilibrium? In their homeworks and exams, students must write extensively about the given differential equations. Essays complete with pictures and numerical data are expected. The students must also go to the computer lab at least 5 times per term and perform elaborate numerical experiments on differential equations. As is their science classes, they must write up their findings, in English, and again with charts and graphs.
In general, this is a vastly different course form the old differential equations course. Interestingly, a very different type of student tends to succeed in this course. The students who are good at algebra alone tend not to do well in or enjoy the course. But the students who may miss a couple of derivatives here and there, but who are capable of much deeper thought about the problem at hand, succeed admirably.
My sense is that, when the appropriate solvers reach the secondary classroom, the experience we have had with differential equations will translate nicely into that setting.