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Spreadsheets in Education (eJSiE)

|Issue 1Volume 1 Article 2

7-24-2007

Spreadsheets in Education –The First 25 Years

John Baker

Natural Maths, john@naturalmaths.com.au

Stephen J. Sugden

Bond University, ssugden@bond.edu.au

This Regular Article is brought to you by the Faculty of Business at ePublications@bond. It has been accepted for inclusion in Spreadsheets in

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Recommended Citation

Baker, John and Sugden, Stephen J. (2003) "Spreadsheets in Education –The First 25 Years," Spreadsheets in Education (eJSiE): Vol. 1:

Iss. 1, Article 2.

Available at: http://epublications.bond.edu.au/ejsie/vol1/iss1/2

Spreadsheets in Education –The First 25 Years

Abstract

Spreadsheets made their first appearance for personal computers in 1979 in the form of VisiCalc, an

application designed to help with accounting tasks. Since that time, the diversity of applications of the

spreadsheet program is evidenced by its continual reappearance in scholarly journals. Nowhere is its

application becoming more marked than in the field of education. From primary to tertiary levels, the

spreadsheet is gradually increasing in its importance as a tool for teaching and learning. By way of an

introduction to the new electronic journal Spreadsheets in Education, the editors have compiled this

overview of the use of spreadsheets in education. The aim is to provide a comprehensive bibliography and

springboard from which others may develop their own applications and reports on educational applications

of spreadsheets. For despite its rising popularity, the spreadsheet has still a long way to go before becoming a

universal tool for teaching and learning, and many opportunities for its application have yet to be explored.

The basic paradigm of an array of rows-and-columns with automatic update and display of results has been

extended with libraries of mathematical and statistical functions, versatile graphing and charting facilities,

powerful add-ins such as Microsoft Excel’s Solver, attractive and highlyfunctional graphical user interfaces,

and the ability to write custom code in languages such as Microsoft’s Visual Basic for Applications. It is

difficult to believe that Bricklin, the original creator of VisiCalc could have imagined the modern form of the

now ubiquitous spreadsheet program. But the basic idea of the electronic spreadsheet has stood the test of

time; indeed it is nowadays an indispensable item of software, not only in business and in the home, but also

in academe. This paper briefly examines the history of the spreadsheet, then goes on to give a survey of major

books, papers and conference presentations over the past 25 years, all in the area of educational applications

of spreadsheets.

Keywords

Spreadsheet, education, mathematics education, statistics education, survey.

This regular article is available in Spreadsheets in Education (eJSiE): http://epublications.bond.edu.au/ejsie/vol1/iss1/2

Spreadsheets in Education–The First 25 Years

John E Baker

Director, Natural Maths

john@naturalmaths.com.au

Stephen J Sugden

School of Information Technology, Bond University

ssugden@bond.edu.au

July 24, 2003

Communicated by S. Abramovich.

Submitted June 2003; revised and accepted July 2003.

Abstract

Spreadsheets made their ﬁrst appearance for personal computers in 1979 in the form of VisiCalc

[45], an application designed to help with accounting tasks. Since that time, the diversity of

applications of the spreadsheet program is evidenced by its continual reappearance in scholarly

journals. Nowhere is its application becoming more marked than in the ﬁeld of education. From

primary to tertiary levels, the spreadsheet is gradually increasing in its importance as a tool for

teaching and learning. By way of an introduction to the new electronic journal Spreadsheets in

Education, the editors have compiled this overview of the use of spreadsheets in education. The

aim is to provide a comprehensive bibliography and springboard from which others may develop

their own applications and reports on educational applications of spreadsheets. For despite its

rising popularity, the spreadsheet has still a long way to go before becoming a universal tool for

teaching and learning, and many opportunities for its application have yet to be explored.

The basic paradigm of an array of rows-and-columns with automatic update and display of results

has been extended with libraries of mathematical and statistical functions, versatile graphing

and charting facilities, powerful add-ins such as Microsoft Excel’s Solver, attractive and highly-

functional graphical user interfaces, and the ability to write custom code in languages such as

Microsoft’s Visual Basic for Applications. It is diﬃcult to believe that Bricklin, the original

creator of VisiCalc could have imagined the modern form of the now ubiquitous spreadsheet

program. But the basic idea of the electronic spreadsheet has stood the test of time; indeed it

is nowadays an indispensable item of software, not only in business and in the home, but also

in academe. This paper brieﬂy examines the history of the spreadsheet, then goes on to give a

survey of major books, papers and conference presentations over the past 25 years, all in the area

of educational applications of spreadsheets.

Keywords: Spreadsheet, education, mathematics education, statistics education, survey.

1ABriefHistory

Paper-based spreadsheets have been around for centuries, primarily for use as record-keeping and

accounting tools. The ﬁrst electronic spreadsheet, VisiCalc [45], [36] appeared in 1979, created by

Dan Bricklin (concept) and Bob Frankston (programmer) for the Apple II platform. It was conceived

and developed as a tool to do repetitive calculations for Bricklin’s studies at Harvard Business School.

They formed a company, Software Arts, to market the product. In 1981, Bricklin received the Grace

Murray Hopper Award from the Association for Computing Machinery (ACM) for the creation of

VisiCalc. It has been said that VisiCalc was the application, more than any other that sold millions

of Apple II computers. Houghton [87] notes:

The invention of the spreadsheet made personal computers have real value in the marketplace

and legitimated the personal computer industry. Without the invention of this software category,

spreadsheets, the impact of the personal computer might have been delayed for years.

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Baker and Sugden: Spreadsheets in Education

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J. Baker and S. Sugden

Bricklin and Frankston sold the rights in VisiCalc to Lotus Development Corporation, which

developed Lotus 1-2-3, a huge killer application for the new IBM PC in 1982. This product introduced

rudimentary database and graphics functionality into the spreadsheet domain, and it dominated

the market for most of the 1980s. Then came Microsoft Excel, which was developed ﬁrst for the

Apple Macintosh, but which was also the ﬁrst real Microsoft Windows application. In fact, the

earliest versions of Excel ran under MS-DOS, but with a special Windows runtime environment [193].

Other major spreadsheets were SuperCalc (1980, for CP/M operating system), Multiplan (Microsoft),

PlanPerfect (WordPerfect Corp.), Quattro Pro (Borland), VP-PLANNER and AsEasyAs. Since the

mid-1990s, Microsoft has held the dominant market share, and now commands in excess of 90% of

the spreadsheet market [193].

The basic paradigm of an array of rows-and-columns with automatic update and display of re-

sults has been extended with libraries of mathematical and statistical functions, versatile graphing

and charting facilities, powerful add-ins such as Microsoft r

°Excel’s Solver, attractive and highly-

functional graphical user interfaces, and the ability to write custom code in languages such as Mi-

crosoft’s Visual Basic for Applications r

°.

Useful summaries of spreadsheet history are given by Power [154] and Walkenbach [194] and

the authors gratefully acknowledge these sources in preparing the present very brief section on the

historical background and emergence of electronic spreadsheets. Dan Bricklin’s own website [45] has

some very interesting titbits of information and, at the time of writing, also includes a downloadable

copy of VisiCalc. Bob Frankston’s website [71] is also interesting.

2 Programming vs Spreadsheets

As early as 1984, just one year after Lotus 1-2-3 made its presence felt in the commercial market,

educators were beginning to discuss their experiences with using spreadsheets in education [25], [26].

Hsiao [88] makes the point that while computers are clearly useful tools for education generally, one

of the main disadvantages is having to program them. In many cases, (at least in 1985), students

had to learn a programming language in order to beneﬁt from computers. Hsiao observes that use

of spreadsheets helps to get around this problem. This view is still supported by other writers; for

example, Morishita et al [133] state that:

Our experience in computing was that it took a very long time to learn computer languages

and it was sometimes very hard to obtain proper results in a limited time. The spreadsheet,

however, is rather easy to use and almost instantaneous numerical simulations are possible.

Hsiao considers examples such as function tabulation and plotting, integration by Simpson’s rule,

and makes a brief mention of the possibility of matrix computations, including the Gauss-Jordan and

inverse matrix methods of solving linear algebraic systems. Although this is quite an old paper, it is

clear that the potential for educational uses of spreadsheets wherever mathematics is involved was

obvious to many people even in the early 1980s. The author makes some interesting closing remarks.

1. The ﬁrst is that the spreadsheet has allowed teachers to adopt a middle course, compared to

the extremes of fully coding an algorithm in some programming language such as BASIC or

Pascal, or using an oﬀ-the-shelf package with a canned solution. It is argued that neither of

these methods is ideal for learning and the spreadsheet approach is recommended.1

2. The second one concerns the absence (in 1985) of a number of desirable features, such as

transposition of rows and columns, absence of regression lines on scatter graphs, 3D graphs, and

so on; all of these features are available in modern versions of spreadsheets such as Microsoft’s

Excel.

1In many instances, at least, the present authors are in agreement.

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Essentially the same points are made in a more recent paper by Steward [176]:

I would suggest that when both are possible, students ﬁnd it easier and quicker to use a

spreadsheet than write a computer program. Moreover, once written a program can often mask

the mathematics that it is intended to represent, while on a spreadsheet the procedure is constantly

exposed.

In this paper [176], the author considers three examples illustrating his claim: linear regression

from ﬁrst principles, convergence of recursive sequences, and ﬁrst-order ordinary diﬀerential equa-

tions.

Relf and Almeida [157] give the example of the spreadsheet being used to (comprehensively) solve

the Birthday Problem, which can be stated as ﬁnding the least number of people in a group for which

there is a probability ≥0.5 that two will have the same birthday. In comparing the cost/beneﬁts of

programming vs spreadsheets, they say:

...theprimeneedis foramediumwhich will facilitate consideration of conceptual issues while

requiring minimal technical expertise, will provide insights into the mathematical context without

necessitating attention to extraneous distractions, will permit modiﬁcation without the need for

major changes in design and will ﬂag and encourage the pursuit of connected enquiries.

It is the spreadsheet that meets all of the above.

With access to VBA (Visual Basic for Applications), educators are also ﬁnding great beneﬁtin

tying programming to spreadsheet use, thereby overcoming much of the time spent in organizing data

input and output. For example, Martin [121] outlines a course for Operations Research students in

which he reports that:

...using the spreadsheet as a platform, the student is led to a position where they can write

virtually a stand-alone program to support a simple OR application, even from a position of little

previous spreadsheet or computing experience.

From the above, it appears that if the goal is not to teach a programming language, but to achieve

understanding of a concept/topic for which some sort of program is needed to show, for example,

variation over time, then the spreadsheet should be the ﬁrst choice.

3 Packages vs Spreadsheets

The Association for Educational Communications and Technology (AECT) has a useful website, and,

in particular, an interesting page [201] entitled Spreadsheets as cognitive tools. On it, we read the

following quotation from the doctoral dissertation of Leon Argyla [109]:

Spreadsheets are powerful problem-solving tools. However, the diﬃculty in using spreadsheets

for problem solving depends on the amount of abstractness and information processing the problem

contains (Leon-Argyla, 1988).

At the same time as spreadsheets, a number of computer packages have been developed that

parallel business and educational applications. In the arenas of statistics and geometry, sites such as

CTI2[202] and CIGS3[203] show that specially designed packages are preferred to the computational

and graphical features of spreadsheets.

However, on the beneﬁt side, we would suggest that:

2ComputersinTeachingInitiative

3Corner for Interactive Geometry Software

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1. Building spreadsheets requires abstract reasoning by the learner.

2. Spreadsheets are rule-using tools that require that users become rule-makers (Vockell and van

Deusen [192]).

3. Spreadsheets promote more open-ended investigations, problem-oriented activities, and active

learning by students (Beare [31]).

Feicht [67] describes the use of Excel to investigate relationships between matrices and geomet-

ric transformations, and in doing so, opens up a very rich ﬁeld for further investigation. Today’s

mathematical syllabus has little time for geometry, but it may see a resurgence if it can be shown

that electronic versions of theorems have the power to excite the imagination and enthusiasm of

learners–see the mathematics site of Bogomolny [204], which is testimony to the power of animating

concepts in mathematics and statistics. For an example hosted in Microsoft Excel, consider the work

of Staples [175]. We have seen that when it comes to programming, spreadsheets have desirable

features; it may be that in the future, the animation of concepts is an area in which spreadsheets

will have an increasingly inﬂuential role over computer packages.

4MathematicsEducation

For applications of spreadsheets in mathematics education, especially that of mathematics educators,

Abramovich, along with co-workers, is the most proliﬁc author ([1]—[22], [110]). Indeed, Abramovich

and co-workers are leading the ﬁeld in this area. The website [1] for Abramovich’s course “Using

Spreadsheets in Teaching School Mathematics” contains a number of useful links, and is well-worth

investigating. Abramovich and Brantlinger [11] examine the suitability of spreadsheets for mathe-

matics teacher education. They note that the typical classroom is not well-equipped when it comes

to a range of mathematical software.

The appearance of computers as alternative instructional tools has created serious problems in

undergraduate pedagogy. This is particularly the case for mathematics teacher education, because

technology as a mathematical/pedagogical tool is not always used appropriately in schools. Indeed,

some teachers who attempt to incorporate technology into the curriculum limit its use to routine

computations only due to a lack of experience with this technology. The mathematics education

community views this problem as a great challenge to educational reform [11].

In [10], the same authors give examples of the use of recursion in a spreadsheet, the visualization

of the limit of a sequence, and cobweb diagrams. The paper concludes that, given the wide avail-

ability of spreadsheets, a toolkit of computer resources (such as a scientiﬁc calculator, a graphing

package and a database program) can be just one program: a modern spreadsheet. Abramovich and

Norton [15] extend this work to investigation of chaos, and associated basic concepts of convergence,

divergence, cycling and period doubling. They further argue that for mathematics education majors,

the spreadsheet is the ideal vehicle for the illustration of mathematical fundamentals such as these.

Similar topics are covered in another paper by Abramovich et al [13].

4.1 Collaborative Constructivist Learning

D’Souza and Wood [200] investigate the beneﬁts of spreadsheets in a collaborative learning environ-

ment at secondary school level, with the application being elementary ﬁnancial mathematics. The

authors state that:

Spreadsheets have enormous potential for assisting in the learning of algebraic concepts. They

can be of great beneﬁt at all levels. Spreadsheets enable students to concentrate on thinking about

the subject matter at hand rather than on the software. There are many mathematical applications

of spreadsheets as noted by Beare [32].

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It is worth repeating the relevant citation from Beare [32]:

Spreadsheets.... have a number of very signiﬁcant beneﬁtsmanyofwhichshouldnowbe

apparent. Firstly they facilitate a variety of learning styles which can be characterised by the

terms: open-ended, problem-oriented, constructivist, investigative, discovery oriented, active and

student-centred. In addition they oﬀer the following additional beneﬁts: they are interactive; they

give immediate feedback to changing data or formulae; they enable data, formulae and graphical

output to be available on the screen at once; they give students a large measure of control and

ownership over their learning; and they can solve complex problems and handle large amounts of

data without any need for programming.

Herrington and Standen [84] consider that many multimedia educational packages tend to present

material in an instructivist manner, thus placing the learner in a passive role. They would prefer to

see learning posed in an authentic setting to provide a constructivist learning environment, and it is

the spreadsheet that provides just such an environment4.

In a 1997 conference presentation, Sher [165] asserts that:

The spreadsheet is the ideal environment for software that follows the Harvard ap-

proach.

In a nutshell, the so-called Harvard approach [for mathematics teaching] is that every topic should

be presented geometrically, numerically, and algebraically. Such an approach is exempliﬁed in the

work of Hughes-Hallett [89] and many others. Similarly, Friedlander [72] suggests that:

Spreadsheets build an ideal bridge between arithmetic and algebra and allow the student free

movement between the two worlds. Students look for patterns, construct algebraic expressions,

generalize concepts, justify conjectures, and establish the equivalence of two models as intrinsic

and meaningful needs rather than as arbitrary requirements posed by the teacher.

Thus, in mathematics education, we are seeing the emergence of the spreadsheet as a key compo-

nent of the constructivist teaching repertoire. There is no doubt that this role will grow in importance,

as students become more and more competent in the mechanics of using spreadsheets.

We close this section with some comments from a relatively modern work, that of Ainley et al

[24]:

Over recent years, we have explored the conjecture that particular pedagogical settings that

exploit immediate and continuous access to computers can change the way in which knowledge

about graphs is constructed.

...the spreadsheet provides the facilities that allowed Clara and Colin to construct these mean-

ings for trend ...

...Through activities with spreadsheets in this pedagogic setting, children expressed new mean-

ings in which scattergraphs became increasingly powerful tools for analysing ongoing experiments.

4.2 K-12 mathematics

Lewis [114] notes that:

Spreadsheet assignments oﬀer concrete ways to explore abstract concepts in mathematics and

other subjects.

4We are not implying here that spreadsheets are the silver bullet that will transform traditional classrooms into

constructivist ones. Indeed, it is quite possible to imagine students being taught to program spreadsheets in teacher-

centred ways.

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Lewis [115] has also produced a spreadsheet resource book for mathematics teachers of grades

K-8. Nowhere is the exploration of an abstract concept more fully researched than in the study of

Sutherland and Rojano [184] in which the potential of the spreadsheet to enable students to form a

correct understanding of algebraic concepts such as pronumerals is investigated. This frequently cited

work, in a sense, sets the benchmark for educational studies of spreadsheet potential. Sutherland

[187] has much more recently used the spreadsheet environment to allow secondary school students

in the UK to develop basic concepts of algebraic dependency. Since the students have trouble with

the abstract nature of algebra, the spreadsheet is used to develop relationships with point-and-click.

Note the following remarks from Sutherland’s paper [187]:

The more traditional approach to teaching algebra often involved imposing an algebraic

method with an over-emphasis on the manipulation of symbols and with no acknowledgement

of the value of the pupils’ own approaches. This way of teaching algebra was only successful for

the minority of pupils.

Our experimental work has shown that most pupils do not spontaneously engage in the alge-

braic processes of expressing generality, acknowledging and manipulating the unknown, focusing

on structure, using an analytic algebraic method.

One way to help pupils move from a non-algebraic to an algebraic approach can be through

work with spreadsheets. . . ..Pupils use the mouse to support the expression of general relationships

and to move from thinking with situation-based to more abstract algebraic objects. . . ..Mouse

pointing becomes a way of supporting pupils to express general relationships, which are then

represented automatically in spreadsheet code. Pupils become aware of this spreadsheet code

without explicit instruction and interact with it when they need to modify their constructions.

They begin to use the spreadsheet code in their talk when communicating with their partner and

can write it down when communicating with others. In this way the algebra-like spreadsheet code

is learned eﬀortlessly without explicit teaching. Pupils use the spreadsheet speciﬁc calculations

to help in the construction of general rules and often verify their general rule with reference to

speciﬁc numbers. In this way links between symbols and general numbers are established.

Similar comments are made by Abramovich [12]. Referring to a spreadsheet model to support an

inductive proof, it is stated that the model:

...allowsforthevisualizationofaninductive proof of combinatorial identity, and it cognitively

supports a transition from computing to a formal language of mathematics.

Ruthven & Hennessy [162] give an account of the use of various forms of computer technology and

software to support mathematics learning in seven English secondary schools, in the year 2000. In

this rather lengthy paper, although there is much interesting material, we generally conﬁne ourselves

here to aspects relevant to spreadsheets. Some of the major points are:

1. Students are nowadays more familiar with basic IT applications, including spreadsheets, so that

much less time is spent on trivial matters in Excel, and the class can spend most of its time on

mathematical activities within the spreadsheet. This is in contrast to the graphics calculator,

which is not as ubiquitous as the spreadsheet, and is typically only used in mathematics classes,

whereas the spreadsheet could be used in IT skills courses, business, economics, etc.; it is a

more generic tool.

2. Teacher and students have more control over spreadsheet models, e.g., sequences, than for other

software.

3. In a comparison of classes of software used for mathematics instruction and modelling in the

seven schools (courseware, graphware, Logo, spreadsheet), the spreadsheet was the only one in

use at all the schools.

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4. A teacher at one school had his year 11 students doing statistical text processing in Excel:

some text from newspaper website was transferredintoExcel,thenanalyzedstatistically,in a

short space of time.

In hindsight it seems obvious, but probably one of the most profound, clear beneﬁts of using

spreadsheets that emerges from this study is just that of saving time. The time gained can then be

spent on investigating properties of the mathematical objects created in the spreadsheet environment:

the so-called what-if scenarios. There is huge scope for investigation of dependence on parameters in

almost any spreadsheet model of a mathematical process. This includes not only the “traditional”

mathematics that we might classify under the headings of algebra, calculus, trigonometry, geometry

etc., but also more modern and discrete areas such as string processing, language theory, automata

theory, and of course, combinatorics and recurrence relations. Note the following comments from a

teacher of years 7 and 8:

We’ve used spreadsheets in Year 7 and 8, to enable them to look at handling data, because

they can quickly get tables and produce charts that are much better quality than those that they

can produce themselves. I’ve got the bottom set in Year 7 and it can take them the whole lesson

to draw a bar chart. So it’s particularly successful from that point of view because they don’t

have to draw all the axes so much, and it doesn’t take them so long to develop the ideas because

they’re not having to spend a whole lesson drawing something. They can draw twenty graphs in

a lesson and actually see connections, rather than spend twenty minutes drawing the axes and

then twenty minutes talking and then twenty minutes drawing all the graph.

It saves a lot of time as well with the Further Maths and the graphing that we did. It would

have taken forever to actually plot all the points and see what happens when you transform certain

shapes. Whereas it was done in a ﬂash and they could see and they learnt an awful lot. So then

they were ready and they’d accepted it because they’d seen it happening... Whereas it would have

taken many lessons if we’d actually plotted all these graphs, they’d have just got bored by it. So

that deﬁnitely helped, just kept the pace going. [162]

It needs to be added that, despite the growing number of studies that show positive results for

the use of spreadsheets, curriculum goals such as those produced by State and National Education

Departments place less emphasis on the role of spreadsheets than on that of calculators as a tool for

social constructivism in mathematics. There is little doubt that the main reason for this emphasis

on calculators is based on educational administrators being constrained by the number of computers

that are available in any given classroom. As a referee of this paper has noted: “it is not that

graphing calculators are a superior tool it is rather that graphing calculators are a more accessible

tool.” However, curriculum administrators are beginning to acknowledge the enormous potential of

the modern spreadsheet program to enhance learning opportunities.

For a review of the K-12 uptake of spreadsheets, the ERIC digests by Ozgun-Koca [145] for

the years 2000 and 2001 make a good starting point, as they contain numerous links to web-based

resources. The book [101] is a good source of spreadsheet examples for grades 3 to 8. See also the

work of Dugdale [59].

4.3 Number Theory

In spite of number theorist G.H. Hardy’s claim [82] that “I have never done anything useful”, the

utility of number theory nowadays is undeniable, with prime applications being mathematical cryp-

tography and cryptology. Modern CS/IT graduates need to know at least the basics of cryptographic

computer security systems, and the use of spreadsheets in particular for the illustration of number-

theoretic concepts appears to be on the rise. Just a few examples are the work of Abramovich and

co-workers, cited earlier, and that of Sugden [182]. Sugden uses Excel for illustration of Euler’s

ϕ(n)(totient) function, τ(n)(number of divisors), the Möbius function µ(n),aswellasthebasic

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operations of modular arithmetic, including modular inverse, modular exponentiation, leading to a

32-bit implementation of the RSA public key cryptosystem in Excel.

Abramovich [7] also considers the use of Excel for some basic investigations into number theory,

and gives an implementation of the Sieve of Eratosthenes using a few very simple formulas. He goes

on to consider gaps between primes, primes in arithmetic progression, even showing how to tabulate

and graph the function π(x)(numberofprimeslessthanx), all in Excel. His formula for this latter

task is very long but consists of a sequence of very similar sub-formulas, and one suspects that this

could be simpliﬁed somewhat, given the modern facilities of, for example, Microsoft Excel XP. In his

article, Abramovich quotes earlier authors MacKinnon [119]:

...thebest use of computers in mathematics education is to runprofessional software written

for real purposes.

and Steward [176] (op cit, p20).In a later paper, Abramovich and Brantlinger [11] present further

topics in elementary number theory within the spreadsheet environment. These include Pythagorean

triads, Euclidean algorithm, Bride’s Chair, and the representation of integers as the sum of perfect

powers.

4.4 Combinatorics

Neuwirth [135] showed that:

...spreadsheets as a model for mathematical relations can help gaining insight into recursive

relations for combinatorial formulas.

This paper and Neuwirth [136] use the very fundamental features of a spreadsheet to illustrate

the structure of elementary recursive combinatorial identities, introducing an arrow diagram notation

to demonstrate the concept:

We see immediately that in this graphical notation the formula is the same for any case in a

very intuitive sense.

Neuwirth aims to avoid algebra as much as possible as the students have trouble with this and

with the closely related concept of mathematical induction. Sugden [179] also uses Excel to assist

with investigations into mathematical induction and recursive implementations of arithmetic and

geometric sequences (directly applicable to simple and compound interest respectively). The concept

of a recursive relation is one that lies at the heart of Microsoft Excel’s Fill Handle implementation,

which takes a range of cells in which formulae apply and spreads them over a range in a way that

replicates the recursive process. We believe that an investigation into student understanding of

recursion can be greatly enhanced by implementing such a process on the spreadsheet. For example,

Sugden [179] uses this technique for his discrete and business mathematics classes to implement a

full superannuation model, complete with rollover and schedule of net worth, in Microsoft Excel,

with just one addition and one multiplication–no exponentials or intrinsic functions. Given such an

environment, the traditional conceptual diﬃculty of recursion is replaced by the visual simplicity of

a single-step formula (one recursive step) and Excel’s ﬁll-down. Hvorecký and Trencansky [94] use

cell insertion in Excel, under VBA/macro control, to investigate graphs of functions of one and two

variables, and to the generation of fractals. As further cells are inserted (their requirement for extra

cells grows exponentially), graphs and fractals of higher precision are obtained.

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4.5 Numerical Analysis

There is a healthy literature on the uses of spreadsheets in numerical analysis (NA). A topical example

would be Lawson and Tabor [108], in which a model arithmetic and geometric growth is presented

using a spreadsheet. Their application is to model the spread of Variant Creutzfeldt-Jakob disease

(mad-cow disease, or vCJD). In all references located, the positive aspects of spreadsheet use recur.

For example, Soper and Lee [178] conclude:

The versatility of computer spreadsheets makes them a very suitable means of accurately

undertaking numerical calculations. The formulae required are easily entered into the spreadsheet,

and the approach has the further advantage that the users can see what is being calculated, so

enhancing their understanding of the method. With the availability of graphical facilities, a plot

of the value of a variable at successive iterations can easily be displayed. A further beneﬁtof

the approach is that students become proﬁcient in the use of a very popular business computer

package.

Another two examples of using spreadsheets for numerical analysis are from Australia: Smith

[174] describes an entire NA course at Monash University, Australia (1989) taught entirely using the

spreadsheet program VP-PLANNER, and McLaren’s book [126] has been used to support his NA

course at LaTrobe University.

Sequences, a key element of numerical modelling, are a natural for a program such as Microsoft

Excel. There is no doubt that the modelling of sequences is one of the great strengths of the modern

spreadsheet program; in fact, even the very early versions such as VisiCalc were almost as good.

Sequences can be deﬁned by just listing the terms, by a direct formula, or by a recurrence. For the

most basic of sequences (arithmetic and geometric), the most natural deﬁnition is the recursive one:

start anywhere and either keep adding a constant or keep multiplying by a constant. From the point

of view of student understanding, such a prescription is almost too easy to implement in a modern

spreadsheet such as Microsoft Excel.

For the 21st century, IT-literate student, it must seem like a total a waste of time that the teacher

has to write a recurrence relation to express such a simple operation as ﬁll-down ! Students familiar

with spreadsheets already know all about this. The challenge for the teacher is, of course, to relate

the mathematical formalism of a recurrence relation to the readily comprehended spreadsheet notion

of ﬁll-down. Student reaction to the translation of mathematical formalisms such as the algebraic

expression of a recurrence relation such as that of Equation 1 into the spreadsheet environment are

discussed brieﬂy by Sugden [179]. Equation 1 represents a superannuation model with $100,000

rollover (initial value), 1% interest per month, and $500 contribution per month. It is interesting

that many students have diﬃculty even understanding Equation 1 (let alone solving it), yet have

very little trouble implementing the corresponding model in Microsoft Excel. Clearly, there is ample

scope for future research here.

an=½1.01an−1+ 500 if n>0

100,000 if n=0 (1)

It is clear that immediate application of such recursion is to be found in the area of elementary

mathematics of ﬁnance. Coupled recurrence relations (diﬀerence equations) relating to population

dynamics, for example, are also easily handled. Since many sequences are most naturally deﬁned

recursively (arithmetic, geometric, Fibonacci etc.), the spreadsheet oﬀers a very rich environment for

investigation of sequences, in most instances, with no coding required. Further discussion may be

found in [180].

4.6 Mathematics not otherwise cited

Due to limitations of time, it was not possible to obtain a copy of every publication considered

relevant to the application of spreadsheets to mathematics teaching and learning. However, in the

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interests of providing as complete a bibliography as possible in the present survey paper, the following

additional works are cited: [30], [41], [58], [60], [61], [62], [63], [85], [86], [93], [96], [97], [117], [118],

[120], [122], [127], [125], [132], [138], [139], [142], [143], [148], [151], [152], [158], [159], [161], [164],

[172], [173], [185], [190], [191], [195], [197] and [199].

5 Statistics

Does the spreadsheet have advantages that make it a more acceptable tool for teaching and learning

than the use of statistical packages? In this respect, Hunt [91] refers to teacher belief as follows:

Many teachers now believe that a spreadsheet provides a better educational environment

in which to teach statistics at an elementary level. The spreadsheet can be made much more

transparent to the student, allowing them to look inside the black box.

Overall, however, the jury is still out on this question–at least at the tertiary level. Although

Neuwirth [137], for example, has demonstrated that spreadsheets may be employed to assist statistical

learning beyond just that of using intrinsic functions (he writes about visualizing correlation), there

is a good deal of negativity in the literature when it comes to statistical computations. For example,

writing about versions of Excel 5.0 and earlier, Nash and Quon [134] conclude that:

Spreadsheet vendors must be encouraged to do better. Closer attention to statistical issues

would result in tools better suited to data exploration and analysis, and cleaner software design

would avoid some obvious sources of error.

They would like to see options for:

1. Histograms, possibly with unequal class intervals.

2. Stem and leaf diagrams.

3. Boxplots, especially with multiple boxplots on the same scale.

4. Quality-control charts.

5. p−pand q−qplots for distributions, especially the Gaussian distribution.

If the requirement, even at tertiary level, is to cover only elementary concepts, others ﬁnd that the

spreadsheet serves admirably. For example, in teaching an engineering subject, Hall [81] concludes

that:

This approach to teaching mathematics, with an intimate mix of mathematical theory, nu-

merical examples and graphical representations, together with the use of modern computing aids,

seems appropriate for engineers.

Warner and Meehan [196] express similar support, stating that:

Individual instructors will need to weigh the costs and beneﬁts of using a spreadsheet program

versus specialized statistical package, but we believe a spreadsheet, such as Excel, will prove more

attractive in many situations.

Hunt [92] reasserts the belief of teachers, concluding that although Microsoft Excel has a great

deal to oﬀer at the elementary level, it is of limited use to serious students of the subject. He points

to features such as NORMDIST(1.96) which returns the tail probability of 0.975 and commends the

function NORMSINV(RAND()) which can be used for simulating a standard normal random deviate.

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Features such as these really do make the spreadsheet a simple-to-use tool for the constructivist

learning environment. Hunt also identiﬁes two types of computer-assisted learning activities that are

well-supported by spreadsheets. There is the ‘do-it-yourself’ type of activity favoured by Callender

and Jackson [49], and the ‘Blue Peter5’ activities where students experiment with pre-set spreadsheets

such as the DISCUS materials of Hunt and Tyrrell [90].

Callender [48] demonstrates how the spreadsheet, with simple-to-write macros, can be used to

demonstrate the normal distributions and the central limit theorem. He suggests that a beneﬁtof

using spreadsheets to introduce such key topics in statistics is:

The normal distribution has been introduced without ‘deﬁnitions’ being needed and all the

components of the Central Limit Theorem are demonstrated as a consequence of probability

sampling.

However, a recurring complaint is that Microsoft Excel does not cater for exploratory data analy-

sis. We would question the value in Excel’s catering for such a speciﬁc and small audience as that

for whom exploratory data analysis is an issue. Features such as box and whisker plots can readily

be constructed on XY-scatter charts, and once the framework for such a plot has been made, it can

be quickly copied for use with diﬀerent data. To make a histogram with uneven intervals, our advice

would be ‘Don’t!’ and if you have data that needs to be summarized on a histogram, the Pivot Table

facility provides a very speedy way of generating the required data table. It would appear that the

true value of these educational oddities of exploratory data analysis have not been recognized by the

wider business community, and hence not implemented by Microsoft programmers.

More serious is the common complaint that Excel statistical functions are unreliable [123], [124],

[104]. Indeed the present authors would hesitate to recommend any version of Microsoft Excel

for professional statistics work, although the second author has done much consulting work based

(stochastic modelling) in Excel [181] and the package performed very well. But, from a pedagogical

standpoint, it is here argued that a collection of totally robust algorithms for computing statistical

functions and distributions is not the principal requirement for students when learning basic statistical

techniques and functions. Provided the implemented functions are reasonably accurate for normal

ranges of the parameters, it is far more important that the student is able to understand the operation

of such functions and so see the many connections, patterns and properties; these are amply illustrated

in the spreadsheet environment. However, a revamp of the statistical functions appears to be on

Microsoft’s wish list for Excel 2003 [193].

In this context, please forgive us for slipping in a positive comment about the MODE function; one

that is taught in very elementary statistics courses, never again to rear its head. In Excel, however,

we ﬁnd that the MODE function is a powerful tool for locating duplicates in a list, particularly when

combined with conditional formatting (e.g., if the value of this cell is =MODE(range) then format

the cell to have a blue colour). Some further useful information concerning the use of conditional

formatting with Excel’s MODE function was pointed out by an anonymous reviewer, and is included

here, with permission.

It appears that in the case of poly-modal array of numbers, the MODE function recognizes the

ﬁrst number that appears most in this array. For example, using conditional formatting jointly

with this function, in the array {2,1,3,2,1,5,5,3,4,4}twos will be highlighted only, whereas in

the array {1,2,3,2,1,5,5,3,4,4}ones will be highlighted only.

It would be inappropriate to complete this section without mentioning the Association of Statistics

Specialists Using Microsoft Excel (ASSUME) [205] whose main interest appears to be the use of Excel

in higher education. Queries to the ASSUME list will almost always receive an answer from someone

who knows.

5This name refers to the English children’s TV program ‘Blue Peter’ catchphrase — “Here is one that I

prepared earlier".

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6 Physical Sciences

Spreadsheets have long been used both for teaching mathematical principles of physical science and

analysis of empirical results; for example [27], [31], [34], [35], [39], [40], [42], [43], [44], [46], [47], [50],

[51], [52], [53], [56], [57], [65], [66], [68], [69], [75], [77], [79], [80], [99], [100], [102], [103], [105], [108],

[111], [128], [129], [130], [131], [133], [140], [141], [144], [149], [153], [163], [169], [188], and [198].

However, in his 1992 paper, Smith [171] notes the comparative rarity of using the spreadsheet as a

mathematical teaching tool. He goes on to mention some uses of spreadsheet in teaching physics, and

cites the texts of Misner & Cooney [130] and Crow [52]. In his closing remarks, Smith predicts three

positive outcomes for students:

1. Reversal of the declining interest in mathematics.

2. Improvement of technological literacy and enhancement of career preparation.

3. Revitalization of mathematical skills through problem solving.

Some textbooks, not mentioned elsewhere, and which employ spreadsheet-based models, algo-

rithms and solutions are [25], [26], [27], [33], [37], [38], [54], [55], [83], [106], [107], [150], [155], [170],

[183], and [186].

An interesting modern paper is that of Lim [116], which describes how spreadsheets are used to

support the teaching of quantum chemistry. It is reported that students with weak mathematical

backgrounds have proﬁted from this approach.

This author prefers the use of spreadsheets for weaker students for the following reasons. The

symbolic mathematical packages depend on the use of a symbolic, quasi-programming language,

which can present an additional learning obstacle for many students....

Furthermore, the access to symbolic mathematical packages is usually more limited than

that of spreadsheets, which are widely available in home, business and community settings. The

“worldware” ...also called “application-software” ... nature of spreadsheets means that students

will have greater opportunities to use and become familiar with spreadsheets than with (eg)

symbolic mathematical packages, leading to greater utility and expertise. Software that isn’t

designed for instruction can still be good for learning....

A “straw poll” of physical chemistry faculty suggests that signiﬁcantly more faculty use spread-

sheets in teaching and learning activities than symbolic mathematical packages [116].

7ComputerScience

Shinners-Kennedy [166] reports on the use of early work in which spreadsheets provide a host envi-

ronment for teaching assembly language programming. Even as early as 1986, he notes:

It is surprising the range of applications that have been coaxed into this format.

In another paper [167], the same author describes computer programming and data structures and

algorithms courses and refers to the well-known obstacles to comprehension of run-time behaviour of

programs for "weaker students". The response of several educators to this problem is the creation of

run-time visualization environments. Similar considerations are expressed by Rautama et al [160].

Once again, the authors state that many students have trouble understanding basic algorithms, and

it is a signiﬁcant challenge to inspire them to investigate why and how a given algorithm works.

Examples they cite are binary search, and edit distance for string editing. Rautama et al use

Microsoft Excel as an algorithm animation environment, and also point out that such an approach

leads to useful tools for algorithm research too. Shinners-Kennedy [167] recommends the use of

spreadsheets for construction of such visualization systems, and goes on to describe the beneﬁts of

the Microsoft Excel Object Library for such endeavours:

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The spreadsheet concept is deceptively simple.

In essence the spreadsheet system is a toolkit (emphasis ours) for exposing and explaining the

principal concepts associated with the object-oriented methodology.

Fone [70] demonstrates the value of using Excel to model neural networks. In an end-of-course

survey, Fone found that:

Unlike previous groups, there were no comments suggesting the need for better models. How-

ever, ﬁve comments typiﬁed by the statement, “I found the Excel example particularly helpful, not

only in helping me understand the networks but with aspects of other modules” were of particular

interest.

Other applications of spreadsheets in computer science are described in [95] (programming),

[112] (arithmetic unit simulation), [113] (automata), [168] (object-oriented programming), [64] (ani-

mation).

8 Economics and Operations Research

Thiriez [189], in an invited review of the role of spreadsheets in operations research teaching begins

by pointing out that there is:

...no point in demonstrating the eﬃciency of the spreadsheet as a tool for teaching basic

modeling and programming.

Thiriez also notes that there are less well-known features of spreadsheets that turn out to be

powerful, so long as you know how to use such features. He goes on to give examples in decision

theory, linear programming and simulation:

The spreadsheet is the best tool for teaching deterministic simulation.

Suggestions are also made that users should view the Solver capabilities with caution. However,

hiskeypointisthat:

A major asset of spreadsheets resides in the speed with which a model may be designed.

Recently, I had the opportunity to build a simulation model for the automatic handling of luggage

at the Paris Roissy airport. When the model (totally developed in Excel) was ﬁnished, I learned

that all other proposals for the tender oﬀer of this application, all of them based on the use of

professional simulation software, had all been at least ﬁvetimesasexpensiveasourproposal.

Many other examples of the use of Excel’s Solver are to be found on the WWW; for example, see

[177].

Jones and Judge [98] support the general views of Thiriez in a paper that gives examples from

micro-economics and macro-economics, illustrating how a spreadsheet can be used to assist in the

development of a student’s understanding of dynamic models and their properties. Adams and Kroch

[23] point to the value of the graphical component of spreadsheets:

The linkage between algebraic, numerical and graphical presentations in spreadsheet programs

on the personalcomputeris asignaladvantage for theteaching of macroeconomics...toillustrate

the principal elements of macrotheory.

Goddard et al [78] note the way in which spreadsheet usage supports a shift from active to passive

learning by enabling students to explore what happens to income-expenditure economy models (the

IS-LM model) as equilibrium is disturbed.

Barreto and others use Excel for the teaching of Economics and Econometrics at Wabash College

[29].

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9 Some spreadsheet esoterica

9.1 Conditional formatting

An interesting feature, introduced in Excel 97 is that of conditional formatting. It may be viewed

as a generalization of the common accounting practice of colouring amounts of money red or black,

depending on their natures as debits or credits (negative or positive). Microsoft Excel is able to

automatically format (colour, border, font etc.) a cell based on its current value. A simple example

of the use of this feature is solving f(x)=0without algebra, but by just observing change of sign

(change of colour). Those interested in the use of this facility will ﬁnd downloadable models on

the Spreadsheets in Education (SIE) eJournal site [28]. The literature on the use of this feature for

educational applications within the spreadsheet environment is scant indeed, although applications

abound. One reference to its use is that of Sugden [179], who uses it for solving f(x)=0without

algebra, and illustrating the solution of simultaneous linear congruences, among many others.

9.2 Names

One of the most beneﬁcial features of the modern spreadsheet is the facility to deﬁne a Name. Such

a feature allows the spreadsheet user to refer to a cell or collection of cells by a single identiﬁer; this

is similar to the use of variable names in programming languages rather than hard addresses (relative

or absolute). Many beneﬁts accrue with the use of names; some of these being:

•The troublesome, but sometimes necessary, absolute references are handled automatically.

•Models may be expressed in notation very close to that of standard algebra. Instead of using

an obscure formula with hard cell references such as = A2*$E$3, one may write something like

= GrossPay*TaxRate.

•Large areas of the worksheet such as tables or lists may be easily selected by just going to the

Name box (Microsoft Excel)

•Models become at least partially self documenting and tend to be easier to debug.

Despite the fact that problems with relative and absolute addressing simply vanish, and that

formulasmaybeexpressedintermsverycloseto an algebraic model when one uses names, it is

rather surprising how rarely these are used in published examples.

9.3 Auditing and debugging

In this paper, we have trumpeted the beneﬁts of spreadsheets for educational purposes. What is the

downside? There exist quite a number of papers which highlight negative aspects of spreadsheets,

and in the interests of a more balanced presentation, it seems only fair that our survey should

include some of these papers too. The main objection appears to be debugging (touseacomputer

programmer’s term), or auditing (to use an accounting term). When applied to spreadsheets, these

two terms refer to much the same thing: the problem of ensuring that a given model is correct.

The diﬃculties of auditing spreadsheets are well-known, and the level of complexity seems to be

comparable to that of debugging a moderate-sized computer program; this problem is well-known to

be hard [73], [74], [146], [147], [156].

To put our negative remarks about spreadsheet auditing in perspective, we note that modern

spreadsheets such as Microsoft Excel have some quite useful auditing tools; for example, even without

invoking the auditing toolbar, just a double-click in a cell will highlight immediate antecedents, that

is, cells which directly aﬀect the value of the current cell. Secondly, spreadsheet models developed for

educational purposes tend to be founded on a sound mathematical model, and if wrong results are

produced, or a limitation of the model is reached, then students can be asked to investigate why such

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a happening has occurred. In other words, in an interactive learning environment, it is often possible

to turn an apparently negative outcome to a positive one. For an example of this, see Sugden [179].

10 Conclusion

In this brief outline of spreadsheets in education, one of our principal aims has been to provide

arguments and motivation for further research in this area. There is no longer a need to question

the potential for spreadsheets to enhance the quality and experience of learning that is oﬀered to

students. Traditional barriers (particularly the lack of facilities to use spreadsheets in assessment

contexts) need to be removed, either by ensuring that access to computers is improved or by changing

assessment methods. Further expansion is needed of the types of topics that can be eﬀectively covered

by spreadsheet examples; for example, one of the authors has recently completed a spreadsheet to

enable the investigation of cellular automata as described by Gardner [76].

Hence the electronic journal Spreadsheets in Education! The goals of the journal [28] are:

1. To create a forum in which ideas on the use of spreadsheets can be exposed and explored.

2. To provide an avenue for scholarly research into the use of spreadsheets at all levels of education

to be reported to a receptive, practising audience.

3. To enable teachers to take on board a technology which can rightly claim to be one of the

founding fathers of the personal computer and which is a tool for life to which all students

should be exposed.

The electronic medium was chosen over print so as to provide a facility to discuss spreadsheets

in education and at the same time giving access to the spreadsheets under discussion. In the view of

the authors, to do otherwise would be inappropriate in this age of technology.

Acknowledgment

The authors are grateful to the reviewers for many useful comments and suggestions, and espe-

cially to Sergei Abramovich, State University of New York, for coordinating the review process.

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