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Calculators policies in NIU math courses

In a separate document you will find specific guidelines describing the use of calculators in mathematical sciences courses at NIU. Since there is a variety of opinions about the value of calculators in the mathematics classroom, the development of these guidelines has required careful reflection. Some key ideas informing our decisions are discussed here for the benefit of those interested in the development of pedagogical practice.

Some plusses and minuses of calculator use in the classroom

The variety of calculator requirements has been designed with the best interest of the students in mind. Our goal is to help the students understand and use mathematics, and the department believes a judicious use of technology is most likely to enhance the students' learning. This includes calculators as appropriate tools for faculty to demonstrate, and for students to explore, some of the concepts as they appear. Most exams, and some homework assignments, however, are not designed for exploration; students are expected to demonstrate that they have now mastered the underlying concepts.

An essential ingredient of proper calculator use is the ability to determine when it is appropriate to use a calculator, and when not. It would be unproductive for college students to divide 0.9738 by 0.30103 by hand; the same is not true when dividing 1 by (1/20), since the likelihood of mistyping exceeds the likelihood of getting a wrong answer by hand. (More precisely, if a college student does not already know that 1/(1/x) = x then it is pointless to take any college-level mathematics course.)

Parallel examples exist with the use of symbolic calculators in college-level mathematics classes. Should students, for example, learn to compute partial-fractions decompositions by hand? It is important in calculus to understand the principles of this decomposition to be able to integrate 1/(x2-1), and to work with Laplace transforms in a later differential equations course. So it is certainly appropriate for instructors to expect students to learn in Calculus 2 the steps for constructing such a decomposition. On the other hand, a bit of preliminary effort is enough to convince even the most dogged student that little light is shed on the problem of integrating 1/(x5-1) using Partial Fractions; once the principles are clear, it is reasonable to ask a symbolic calculator to compute the integral.

In both preceding paragraphs, the pedagogical principles are the same. We see that a student who is working needlessly complicated examples by hand is learning little and could be a more productive learner by leaving those routine computations to a machine. At the same time, we see that a student who is working very simple examples by machine is learning little and could be developing a more authentic understanding of the material by trying those computations by hand.

Always keep in mind the fundamentals of the mathematics. Spotted on the internet:

"The purpose of a calculator is to calculate.  The reason they are required 
for math courses is because students were spending so much 
time and effort calculating,  they weren't learning the concepts. 
Now that they've got calculators, they're spending so much 
time and effort calculating,  they  aren't learning the concepts."

Different policies apply to different courses

The importance of working without calculators is especially clear when students are learning ideas which are used in courses for which a given course is a prerequisite. In these courses, it is very important for the student to work large numbers of examples and to be familiar with concepts and procedures, as well as with examples likely to require more computation than can be done by hand. It is in these courses which students are expected to demonstrate mastery of ideas without the use of calculators. Thus in courses with the lowest course numbers (e.g. Math 110), the student should expect not to be able to use any calculator on exams.

On the other extreme are courses primarily designed for applications, that is, terminal math courses. Here it is important for students to be accustomed to using a calculator to work "real" problems, more similar to expected later applications. Thus students are expected to have calculators, especially during exams, for courses such as Math 101 or Math 206.

Intermediate-level courses must be treated with some care. Math 211, for example, is a terminal math class for many; but it can also be used to fulfill a prerequisite for Stat 301, and so students will be expected to understand the principles of the procedures introduced well enough that they can work simple problems by hand. Math 210 is also terminal for many students; however, it is taught and tested in large lecture rooms, and so instructors may for procedural reasons find it necessary to limit or prohibit the use of calculators.

Not all calculators are alike

We attempt to direct the students to the appropriate tool for each job. Four-function calculators are inexpensive and widely available, and make it essentially impossible to derive an undue advantage on an exam; they also offer little useful help towards learning, except to remove the distraction of routine arithmetic.

"Scientific" calculators are capable of more complex numerical functions. Students are expected to be able to use these to compute nontrivial trigonometric values; to compute statistical measures of data; to handle real data in models using logarithm and exponential functions. Some may be programmed, which is usually a useful exercise for students learning algorithms such as numerical integration, but of course overuse of a packaged routine can enable students to avoid learning what numerical integration (and thus integration itself) really means.

"Graphing" calculators produce a pixelated approximation of graphs, curves, and other planar figures. These can be invaluable for students who are investigating properties of functions which are new to them, and are useful for tracing or deriving other data from functions whose overall behaviour students already understand. On the other hand, students who rush to use these prematurely often appear not to see any connection between a function and its graph than memorized associations.

"Symbolic" calculators are capable of algebraic manipulations of functions of one or more variables, and in particular may be able to factor polynomials, differentiate or integrate, and evaluate series and limits. They typically include an alphabetic keypad and may have a text-storage area which of course may be used to store crib notes. Software for computation of symbolic algebra is known to be subject to many errors (some quite sophisticated and interesting!) These calculators typically have a graphic display; some claim a 3-dimensional graphics capability but the displays are usually too primitive to be very helpful.

At present, full-fledged computers are used by students only on their own or in computer labs (i.e. not ordinary classrooms). There is a wide variety of software capable of professional-quality numerical computations, symbolic manipulation, and graphical display, as well as educational and testing software of mixed quality. It is unreasonable to expect most students to be able to use the software without significant direct instruction specific to the use of a computer (i.e., not math per se). At present we do not use these labs in any of our regular math classes. Note that modern computers also double as communications and storage devices, making their use in the classroom problematic.

Exams are special situations

Exams can be high-stress moments, and students often prefer to have a calculator available to check simple arithmetic. While most instructors are perplexed that students could attempt college-level mathematics courses without complete facility in arithmetic and algebra, they recognize that the primary purpose of exams in these courses is not to test elementary skills. When possible, we try to allow use of these, even as crutches, when appropriate.

Perhaps more significantly, we would like during an exam to test the students' ability to use truly representative data. For example, it is clearly preferable to be able to expect trigonometry students to solve triangles with other than "round" angles, and calculus students to compute numerical integrals with at least modest precision. Thus we do ask students to have a calculator available in some of these mid-level service courses.

As with any tool, it is reasonable to expect that considerable practice is necessary to be able to use the calculator optimally. But while it may be optimal for a student in practice to use a machine to carry out a task, it might instead be optimal for the instructor during a course to measure the student's ability to carry out the same task.

In particular, calculator use is not appropriate when students are being tested on their basic conceptual understanding. A common approach fruitfully used by some instructors is to use two-part exams, one part allowing calculators and the other not, as appropriate for the concepts being tested.

Some faculty have also expressed concern that calculators and similar devices provide too much temptation to cheat. The use of a calculator's text-storage feature must be limited during closed-notes exams; the use of a calculator as a communications device must be limited if group-work is forbidden; and the use of programmable calculators must be limited during an exam designed to test understanding of algorithms (e.g. Newton's method). Some instructors successfully delineate ground rules and so may be able to permit appropriate calculator use during exams. Others, particularly when teaching large lecture sections, are unable to police calculator (mis)use, and so may prohibit calculators altogether.

Some instructors have also expressed fear that the use of calculators is detrimental to student success and have expressed great concern for weak students who tend to waste valuable examination time uselessly attempting to query the calculator for an answer which it is not designed to help find. Accordingly, during final examinations in lower-division courses, graphing calculators are allowed only on any specifically-designated portions of two-part exams as described above. These exams will reflect what these instructors are actually doing with their classes, and what they expect the students to understand.

Policy development

These arguments were discussed at length by the Department's Undergraduate Studies Committee during the 2000-2001 and 2001-2002 academic years before this document was made public on the web. (The Director of Undergraduate Studies takes responsibility for the slant of the current wording!)

We continue to experiment with a variety of teaching and testing strategies. To the extent that our expectations of the students' use of technology change, we will of course adjust the final-exam policies.

There is much to be said about the effective use of technology in the classroom and this document can hardly claim to represent all the current research and teaching experience which has been gathered. There is considerable literature and discussion concerning the use of calculators in the mathematics classroom. A (searchable) collection of mailing lists and discussions including educators' groups is available at the Math Forum. Here is a sample of what you'll find. Studies with some degree of professionalism also exist; here is a pointer to some studies, and here is one sample.

Sample calculator errors

Comparison of calculator-permitted and calculator-forbidden Math 230 exams.

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Last modified: 2002/01/25 (djr)