Programmable Calculators
and Mathematics

Barry Kissane

Australian Institute of Education
Murdoch University
Australia


ABSTRACT: Since graphing calculators are generally programmable, recent technological developments have the potential to provide many students with easy access to a programming facility. Some observations are made about the programming constraints of calculators, and conclusions for good practice drawn. Four different levels of use are described. A preliminary analysis suggests that there are at least six different kinds of uses of programmable calculators in mathematics. Some examples of these are given. and their value to the secondary curriculum briefly discussed.


Hand-held programmable calculators have been accessible for about 15 years now, although they have not been used much in schools. In some countries, such as Australia, a major reason for the lack of use has been the difficulty of designing and conducting public examinations fairly. In the last five years or so, graphics calculators have become available and affordable to schools, and are seen to have a number of clear advantages over scientific calculators, and even some advantages over computers. In fact, in the opening session of the calculator miniconference at ICME-7, Bert Waits suggested that we should acknowledge that graphics calculators actually are computers. An interesting feature of graphics calculators is that almost all are programmable, and have enough memory for several programs to be stored for later retrieval.


It seems likely that the bulk of students in secondary schools in industrialised countries will soon have routine access to a graphics calculator, and thus a programmable calculator, as they now do to a scientific calculator. The portability of the machines further suggests that they might be available to students in places that computers are not, such as in examinations and at home. It is surprising then that there is so little literature available, whether analytic, descriptive or research, pertaining to this topic. For example, the NCTM's excellent recent yearbook (Fey & Hirsch, 1992) contains almost no reference to the programmability of calculators. A developmental project for preservice teachers, based at Western Michigan University, is one of the few serious attempts to exploit this tool. (Channell & Flanders 1992). This paper provides an analysis of some of the features of programmable calculators, to assist a consideration of the merits of using them in secondary schools. Because of space limitations, the paper does not address the practical and pedagogical implications, although these would need serious consideration should the idea of using programmable calculators appear to have educational merit.


Although programmable calculators have many similarities to each other, there are also many differences. In this paper, my remarks are based mainly on my experiences with a Texas Instruments TI-81 graphics calculator, a readily available and affordable machine for many students in secondary schools. However, since there are many similarities between modern calculator programming languages, similar analyses might be expected with different calculators in mind.

One-line programming
The most elemental form of programming involves the automated repetition of a single step. On the TI-81, and on other graphics calculators, commands consist of a single line, which is 'executed' by pressing an 'enter' key. Pressing the 'enter' key again repeats the previous command each time, until a different command line is inserted into the calculator. In this way, the same command can be repeated several times in succession. There are some circumstances in elementary mathematics where this is a useful action. For example, if the following command line is entered:

int (6*rand + 1)

each successive pressing of the 'enter' key will execute the command, which in this case will generate a (pseudo-) random integer between 1 and 6, thus simulating a throw of a single die. Similarly, the more complex command

int (6*rand + 1) + int (6*rand + 1)

will simulate the sum of two dice each time the 'enter' key is pressed. Such one-line programs could be used to pedagogical advantage in early probability learning.
Repetition can be used to advantage to evaluate successive terms of a geometric sequence, by also using the facility most graphics calculators have to recall the result of the previous operation from an 'answer' memory. On the TI-81, the answer memory is referred to as Ans so the following command line will act as a one-line program, giving a new value each time the 'enter' key is pressed:

Ans*1.06

Prior to using this command line, it is necessary to give the sequence a starting value, followed by the 'enter' key. The successively pressing the 'enter' key after the command line is entered will effectively give values from a geometric sequence with common ratio 1.06, or successive amounts under annual compound interest growth of 6%.
The idea of repetition lends itself naturally to iterative sequences, such as that generated by the interesting difference equation Xn + 1 = b Xn (1 ­ Xn) for various values of b and with a starting value of X1 in the interval (0,1). The following one-line program will allow successive terms of this sequence to be generated automatically each time 'enter' is pressed (provided the starting value is first entered, followed by 'enter', and provided also that the B memory contains the desired value of b):

B*Ans(1­Ans)

With such a facility, explorations of sequences like this, which in fact can lead to an elementary example of the mathematical idea of chaos, are made easier than before, and students can have a new investigative tool for personal exploration.

Scratchpad programming
Calculator programs usually mimic the kinds of operations that calculators use, since programmable calculators typically access a rather limited set of program commands. The intention of the programming is usually to automate a sequence of calculator operations. In the case of graphics calculators, some of these operations may be rather sophisticated, such as drawing a graph of a function or constructing a regression line. This kind of programming is sometimes called 'scratchpad' programming, perhaps because programs are often scratched out on a pad before being entered into a calculator. The leap from performing a calculation on a calculator to writing a program to do it is often not very large, and so it is likely that someone familiar with their calculator will find the writing of short simple programs easier than they might otherwise expect. In this regard, Ruthven's (1992) comment at the ICME-7 conference, noting that many senior school students in the UK had made spontaneous use of the programmability of their graphics calculators after they had used the calculators for a while.


In addition to mathematical functions, programmable calculators use some input/output commands and some commands to allow for a limited amount of transfer of control, through the use of goto statements, statement labels (to go to) and branching instructions. Unlike algebraic computer languages, higher level programming power is not available on the less sophisticated graphics calculators such as the TI-81. There are no easy equivalents to while, do...until or even for...next statements, although these are beginning to appear on more sophisticated calculators such as the Texas Instruments TI-82 and the Sharp EL-9300. It is quite possible that computer scientists would find the idea of programming like this almost as dangerous as BASIC, in terms of developing poor programming habits and conceptions, as many would be uneasy about the liberal use of goto statements. As well as programming commands, a programmable calculator will also have some mechanism for storing, retrieving and editing programs, akin to a computer operating system.

Influences on programming style
I think of a programmable calculator as a hand-held tool that might be useful for mathematics. The emphasis is on getting a job done, or exploring some mathematical ideas, rather than on developing efficient algorithms or elegant programs ­ although these are often complementary aims. The nature of programmable calculators suggests that different styles of programming from those used for computers ma be appropriate.

As for computer programs, calculator programs can be written at various levels of complexity and with various degrees of attention to style and documentation. Some of the issues affecting programming style are briefly described in this section of the paper and exemplified in the next section.


Audience ­ a program for oneself can often be cryptic to someone else. That doesn't matter if others are not expected to use it. Similarly, it is not necessary to design sophisticated error-trapping routines in a program if only the author will use it. (An example of an error-trapping routine would be a way of detecting that an input to a program for factorising integers was not actually a positive integer.)


Usage ­ a program that is used only once or twice does not need to be too sophisticated. Many programs are probably like this, as they are written to address a need in solving a problem. Even programs that are used repeatedly may use a fairly rudimentary style.


Prompting ­ if a program is stored for later use (even by the author), a few prompts to the user may make it more useable and less error-prone. It is very frustrating, for example, to be presented with a question mark on a calculator display and to have no idea what input is expected. This can happen even to the author, if a program hasn't been used for a while. It is often easier to rewrite a short program than to reinterpret it.


Length ­ the more sophisticated the style and the more helpful the documentation, the longer the whole programming task takes, and hence the less useful it is to do it at all. This suggest that programs are usually short, with most less than a dozen steps long. Indeed, long and complex programs probably have little place on programmable calculators, and are much better written for computers using high-level languages such as Pascal or Fortran or even BASIC. Short programs are necessarily much more transparent than long programs ­ it is much easier for students to understand how they work. As well as educational factors, calculator storage space is also a constraint here, encouraging a preference for shorter programs.

Levels of use
At least four levels of use of programmable calculators are relevant to secondary schools. These are described below in increasing level of sophistication and demands placed upon the user.


Using ­ in which a student merely uses a program that somebody else has written. The main demands on the student are to understand how to run the program, to input information to the program, and to interpret the output. At this level, it is not necessary to know anything of a programming language or even what a program is.


Changing ­ in which a student interacts with a program written by someone else by changing some of the commands. Such changes may be small, such as changing a formula or a branching condition. Studying the effects of a change allows students access to some important mathematical ideas. At this level, students need to know what a program is, how to edit program steps and a little about the programming language concerned.


Demonstrating ­ in which a program is used to demonstrate some aspects of mathematics to a class by means of an overhead projection device. The demonstrator is usually the teacher, and some knowledge of how a program works is probably important for such demonstrations to be valuable. In many cases, both using and changing are involved in demonstrating.


Writing ­ in which students write their own programs, clearly involving a higher level of thinking than does the use, modification, or demonstration of a program written by someone else. At this level, students need a command of parts of the programming language and the mechanisms for creating, modifying, storing and retrieving programs for a particular calculator. Some understanding of programming concepts such as loops and various kinds of input and output are needed as well.

Some examples
To illustrate some of the elements of programming style, below are four versions of a program written for the TI-81 graphics calculator. The purpose of each version is the same: to convert temperatures in degrees Celsius into Fahrenheit. The first version, with title TEMP1, is the shortest and thus the easiest to write. It allows for a single temperature conversion when the program is run.

TEMP1
Input C
32 + 1.8C STO F
Disp F

The next version, TEMP2, loops back to the start, assuming that a series of temperatures are to be converted efficiently ­ a reasonable assumption, since it would be unnecessary to write a program to do the conversion only once. There is no end statement in the program, although the TI-81 allows both a stop and an end instruction. This program, like many others written for calculators, can only be stopped by making an error. It's not elegant, but it works.

TEMP2
Lbl 1
Input C
32 + 1.8C STO F
Disp F
Goto 1

Version TEMP3 uses Disp statements to prompt the user by showing that the first entry is a Celsius temperature and that the output is in Fahrenheit. This is useful if one other than the author is using the program, or the author has forgotten what the program does.

TEMP3
Lbl 1
Disp "CELSIUS"
Input C
32 + 1.8C STO F
Disp "FAHRENHEIT"
Disp F
Goto 1

The final version, TEMP4, asks the user to signal whether they have finished or not, by inputting a 1 to say that there are more conversions to come. The If statement for this executes the following statement, and skips it otherwise.

TEMP4
Lbl 1
Disp "CELSIUS"
Input C
32 + 1.8C STO F
Disp "FAHRENHEIT"
Disp F
Disp "MORE(=1)"
Input M
If M=1
Goto 1
End

TEMP4 is the first (and probably the last!) program with this kind of interrogation I have written for a programmable calculator. It would seem to be quite unnecessary to go to such lengths except to illustrate an idea.

The sample programs given in the next section are not given in various versions, although in practice educational and social circumstances may suggest that some of the elements of programming style would need to be carefully considered.

Some possible uses in school mathematics
Calculator handbooks generally show several programs as illustrations so that calculator owners can learn to write programs. Many, if not most, of the examples given seem not to focus on mathematics; rather, more common applications appear to be in the sciences, in engineering and in business. By considering these, together with programs that I have written for my own use in mathematics, I have made a first attempt at classification of some possible uses for programmable calculators in mathematics in the secondary school:

1. Situations with intrinsic repetition
2. Evaluation of complex formulae
3. Provision of missing calculator features
4. Pedagogical uses
5. Situations involving random simulation
6. Other

This is clearly not yet a very good classification scheme, since the last category is open and since I can think of several programs that fit quite well under more than one heading. In the limited space of this paper, it is not possible to provide more than a few examples to illustrate the categories.

Intrinsic repetition
Some mathematical tasks necessarily involve doing the same thing over and over again, which is by definition a tedious process. These would seem to lend themselves to short programs. Examples include iterative solutions to equations, such as those using the Newton-Raphson method, and the evaluation of successive terms of a sequence, such as the following example, called SEQUENCE.

SEQUENCE
0 STO X
Lbl 1
X+1 STO X
Disp Y1
Pause
Goto 1

In this case, the definition for the sequence is first entered as a function, represented in the program and in the TI-81 as Y1. This is the same procedure normally used before graphing a function.

A program like SEQUENCE may be used in several ways to educational advantage. For example, a young student may explore the kinds of sequences created by linear functions, or an older student may use the program to investigate the apparent convergence of a sequence by looking at successive terms.

Formula evaluation
A constant facility on a calculator will help with some formula evaluation, such as currency exchange rate calculations using a constant multiplication facility. More complex formulae need a short program, such as the temperature conversion programs above. Even rather simple formula can be evaluated efficiently using a programmable calculator, provided that they are to be used often. The example given here, COSRULE, uses the cosine rule to find the size in degrees of the angle in a triangle.

COSRULE
Input A
Input B
Input C
(A2+B2-C2)/2AB STO X
(180/pi)cos-1X STO X
Disp X

Other examples include measurement formulae (such as that for the volume of a sphere, given its radius), geometric formulae (such as that for the angle between two lines of given slope), algebraic formula (such as that for the sum of a given number of terms of a particular arithmetic sequence) and probabilistic formulae (such as binomial probabilities).

The implications for the curriculum here are not clear. In the last generation, we have moved from curricula that expected formulae to be memorised and used by students, to curricula that require students to look up appropriate formula in a book when needed. Is this automation of the evaluation of a formula merely a further step along the same continuum? Could we take advantage of programmable calculators to move a little further towards the justification, meaning and use of mathematical formulae, and a little further away from the tedium of merely evaluating them?

Missing features
Many calculators are missing some desirable features that other calculators have. This will continue to be the case as technological development proceeds. Most of these 'missing' features can be programmed to remedy the deficiency. The following example, NEWTRPSN, finds the roots of a function using the Newton-Raphson procedure. The function is defined as Y1 and the starting value is X.

NEWTRPSN
Lbl 1
X-Y1/NDeriv(Y1,0.0001) STO R
If abs (X-R)<1E-10
Goto 2
R STO X
Goto 1
Lbl 2
Disp "ROOT AT"
Disp R

Other examples in this category might include numerical integration, the numerical solution of equations, approximation of the cumulative density function for the normal distribution, inversion of small (2 x 2) matrices and operations with complex numbers. Programs like these, unlike many others, would probably be kept in memory for a long period, rather than being discarded after one or two uses. As technological help of these kinds becomes readily available to students, some aspects of the curricula will need review. For example, to what extent should we seek to develop computational skills in students that are easily replaced by a machine?

Pedagogical
It is possible to write brief programs that might meet a well-defined teaching purpose, such as allowing students to investigate the relationships between coefficients of a quadratic function and its graph. Such a program might be written by the teacher and then used by her students. The example here of QUADPLOT would not take very long for students to insert into their own calculators, and could then be used for learning purposes.

QUADPLOT
"AX2+BX+C" STO Y1
Lbl 1
Input A
Input B
Input C
DispGraph
Pause
Goto 1

It would not be necessary for students to know how to write a short program like this, since they could merely copy it from a blackboard, and concentrate their attention on their observations. It is not difficult to think of other programs like this one,

Simulation experiments
Random processes can be simulated with short programs, since most programmable calculators include a pseudo-random number generator. The example given here, DICE, simulates the tossing of a fair die N times.

DICE
Input N
1 STO K
Lbl 1
IPart (6Rand+1) STO X
Disp X
IS>(K,N)
Goto 1
Stop

Screen size is a limitation, since the output is limited to a few lines. Longer programs can be written not only to generate random data but also to aggregate the output and to analyse it statistically, but of course, longer programs are less attractive than shorter ones for practical reasons.

Programs like this one allow more opportunity for an experimental approach to the study of chance. This would complement practical classroom activities and more theoretical approaches to elementary probability.

Other
This miscellaneous category could include many examples (which exposes a defect of the classification scheme rather starkly!). For example, the FACTOR program finds the factors of an odd integer, a useful tool to have if investigating aspects of primes and composites. If the program is used repeatedly, all of the factors of a large integer can be found.

FACTOR
Lbl 1
1 STO D
Disp "NUMBER"
Input N
Lbl 2
If (D>ÐN)
Goto 3
D+2 STO D
If (IPart(N/D) not= N/D)
Goto 2
Disp D
N/D STO Q
Disp Q
Goto 1
Lbl 3
Disp "PRIME"

This program is longer than the previous examples, and contains no mechanisms for detecting improper input (such as an even number, a negative integer or a decimal fraction), which would have made it even longer. Such are the crudities of programming on a programmable calculator!

Similarly, programs to convert between number bases might be useful for an investigative task.

Conclusion
Programmable calculators give access to a powerful personal tool for secondary students, and there seems to be a case for considering some ways of exploiting this. This paper has suggested some of the possible kinds of uses that might be examined, as well as offering a perspective on how programming calculators may differ from programming computers. It would surprise me if there were not some situations in which teachers would find some of the programming possibilities of programmable calculators useful, and worth knowing about. A number of different levels of students use have been identified. Whether the time needed to become proficient in programming a calculator is worth the end result for students (or even for teachers) is still an open question, but it is one worth thinking about and one worth studying.

References
Channell, D. & Flanders, J. 1992, Teaching secondary school mathematics with technology, Second Field Trial Edition, Western Michigan University.
Fey, J. & C. Hirsch (eds.) 1992, Calculators in Mathematics Education, Reston, VA, The National Council of Teachers of Mathematics.
Ruthven, K. 1992, Supercalculators in the upper-secondary classroom, paper presented at the International Congress on Mathematical Education, Quebec, Canada.

Barry Kissane
Australian Institute of Education
Murdoch University
Murdoch, Western Australia, 6150
Australia.

This paper is reprinted from:

Kissane, B. (1994). Programmable calculators and mathematics. In Graf, K-D, Malara, N., Zehavi, N. & Ziegenbalg, J. (Eds.) Proceedings of ICME-7 Working Group 17: Technology in the service of the mathematics curriculum: Proceedings of Working Group 17 at ICME-7, Québec 1992
(pp 129-138). Berlin: Freie Universität Berlin.
[http://wwwstaff.murdoch.edu.au/~kissane/icme-7.html]