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.

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/*(x ^{2}-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/(

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."

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.

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 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.

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.

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

- Specific calculator policies for NIU math courses.
- Undergraduate programs in the NIU Department of Mathematical Sciences

Last modified: 2002/01/25 (djr)

Email:qkong@niu.edu