In Technology in Mathematics Teaching: a Bridge between
Teaching and Learning, L. Burton and B. Jaworsky (eds.), Lund:
Chartwell-Bratt, pp. 261-274. ISBN 91 44 60711 3
An analysis of the
relationship between
spreadsheet and algebra
G.Dettori, R.Garuti, E.Lemut, L.Netchitailova
In recent times, the spreadsheet has been suggested as a tool to
teach algebra in intermediate high school. Our a-priori analysis
of the relationship between spreadsheet and algebra shows the
inadequacy of this tool to express the fundamental characteristics
of algebra, that is, the manipulation of algebraic variables and
relations, which make algebra suitable as a formalism for
describing models. However, with the attentive guidance of a
teacher, the spreadsheet can become a useful tool for motivating
the introduction of some concepts of algebra and for reflecting on
different resolution models.
Introduction
In the last years, several studies pointed out both potentialities and
problems related to using a spreadsheet to teach some typical topics
of mathematical curricula in intermediate/high school, such as
algebra, approximate calculus, statistics (see Malara et al. (1992) for
a review). The decision to use the spreadsheet to teach a discipline
(like, for instance, algebra) sometimes seems implicitly determined
by the assumption that an item of software can be successfully used
to teach/learn any topic which is involved in the use of that software.
We consider this point of view arguable, for two reasons; first, this
position does not specify how deep it is necessary to know that topic
to start using the software; second, it does not distinguish between
"learning something about" and "learning the most important aspects
of" a topic. Moreover, as concerns algebra in particular, some
researches (Capponi et al. 1989, Capponi 1989) point out that the
spreadsheet is not really based on algebraic calculus, since the
expressions it uses as formulae, though containing literals which
recall algebraic expressions, do not have algebraic character. The
design of spreadsheet solutions contrasts with the approach of
algebraic ones, in that algebra essentially gives an operative
language to analyse or to manipulate relationships; its aim is not to
perform computations (Booth 1984, Chevallard 1989). Moreover,
the non- algebraic nature of the spreadsheet is the origin of some
difficulties in transferring algebraic competencies when solving
problems of different kinds.
In this paper we are concerned with the relationship between
spreadsheet and algebra. We analyse in which measure the
spreadsheet can really help students to learn algebra, considering not
only a first approach to it, but also, and in particular, its most
characterising aspects. In fact, we think that experimental researches
that propose the spreadsheet as a positive tool to learn algebra limit
their observations to a first approach, that is, to solving simple
problems which are typical of the elementary school algebra
(Sutherland 93, Sutherland et al. 1993). Moreover, the aim of these
researches seems more to see how to use the spreadsheet to compute
the solution of a problem rather than to check if the applied solving
process is of algebraic nature.
Based on the above considerations, we analyse the spreadsheet from
the point of view of learning the main concepts of algebra. This
entails determining what it means to learn algebra, in particular for
students under 16. Our work differs from other researches in this
field in that we emphasise which parts of algebra can be, or can not
be, tackled with the spreadsheet, instead of pointing out how much
algebra is involved in the spreadsheet. Our analysis emphasises that
the spreadsheet can be useful to introduce some elements of algebra,
but that it results inadequate, if not misleading, for a deep learning of
the fundamental aspects of algebra. In our opinion, some limits of the
spreadsheet can be overcome if we do not use it as a tool for solving
problems, but rather consider its underlying resolution model as a
tool for reflecting on higher levels of abstraction, synthesis and
generalisation, under a teacher's guidance.
Our work has the characteristics of an a-priori analysis, but takes
also into consideration both our own classroom experiences and
those mentioned in the referred papers (Chiappini et al. 1991, Dettori
et al. 1993).
School algebra at age 11-16
Before discussing how suitable is the spreadsheet to teach/learn
algebra, we need to define what we mean for algebra and what is
relevant to teach/learn at age 11-16. We are aware that there are
various conceptions of what is algebra (Usiskin 1988) and that
different countries have different school algebra traditions. These
discrepancies can induce different evaluations of the spreadsheet's
influence in teaching/learning algebra. In order to remain as general
as possible, we took into consideration both a classical conception
(like that proposed by O. Terquem in the last century (Chevallard
1989, pg. 36) and a modern view of algebra which emphasises, as a
crucial starting point, the conceptual break with arithmetic
(Chevallard 1984, Chevallard 1989, Cortes et al. 1990). This break is
characterised by entering into a modelling process that changes the
nature of a problem's resolution (from problem to equation, from
equation to equation's solution by producing a formula, from formula
to calculation). In this view, the algebraic resolution of problems
implies the construction and resolution of algebraic equations, and
this requires the ability to perform algebraic transformations.
As concerns school algebra at age 11-16, we consider meaningful:
• to understand what are variables and unknowns;
• to understand the meaning of formulae (e.g. A=LW), equations
(4X=48), identities (sinX=cosX*tanX), properties (n*(1/n)=1),
functions (Y=kX);
• to learn to manipulate algebraic equations and inequalities
according to the rules of literal calculus;
• to learn to apply algebra, that is, to formulate equations which
model problems or classes of problems;
• to learn to apply the algebraic calculus to the demonstration of
simple theorems.
Using a spreadsheet for solving some algebraic
problems
We want first to analyse if the use of the spreadsheet induces that
break between arithmetic and algebra that is at the base of high level
functions characteristic of algebra (such as synthesis, generalisation,
transformation) in relation to its objectives (that is abstraction,
formalisation, modelling). We address this issue by discussing the
resolution of some meaningful school problems. All examples have
been implemented in Excel 2.2a on a Macintosh (A1-style relative
reference system and $A$1 absolute reference system).
Using relations
Let us consider the following problem: We want to distribute 100
books among three persons so that the second one receives four
times the first one, and the third one as the second one plus 10.
Though problems of this kind are often assigned in the introductory
phase of algebra, they are certainly not the most meaningful, since
they can be directly solved using arithmetic, hence missing the point
of the conceptual break between arithmetic and algebra. However,
this problem can be solved algebraically by solving the equation
X+(4X)+(4X+10)=100, where X is the number of books given to the
first person. This equation describes the problem, and, at the same
time, the solution of the problem in implicit form. Transforming it by
means of the rules of algebra we obtain the equation X=10, which
represents the solution explicitly. Using a spreadsheet, we can only
seek a solution of the problem by trial-and-error, that is, by repeating
several times some numerical computations (more precisely, by
computing the value of some formulas), until the required value is
found. Fig. 1 shows the beginning of a schema of solution often used
by students.
Fig. 1 - a) Data and formulas; b) Data and computed values
A numerical result can be found by inserting different values in
column A, either by copying the line or by successively substituting
different values in the cell A2, until the value in the corresponding
cell in column D is 100. The equation X+Y+Z=100 is not explicitly
expressed. The student solves it implicitly by checking, by himself,
the value of the sum in the last column, but he may never become
aware of the equation. This depends on the fact that spreadsheets do
not allow one to express equations, which, on the other hand, are
basic tools of algebra.
This observation leads us to point out a first fundamental
discrepancy between spreadsheet and algebra: the sign of equality
used in spreadsheets is actually the assignment of a computed value
to a cell, while the equal sign in algebra represents a relation. The
inability to write relations in a spreadsheet implies that it is not
possible to use it to completely represent algebraic models. Hence,
the resolution approaches of algebra and spreadsheets are strongly
different: in algebra the solution of a problem is found by formal
manipulation of equations describing it, while with a spreadsheet
successive numerical approximations must be performed until a
numerical solution is reached. This basic discrepancy can even lead
students to misunderstand what is algebra if they are told that, using
a spreadsheet, they are learning algebra.
Synthesising equations
Using a spreadsheet leads students to recognise which elements are
involved in a problem, and to express part of the relationships among
them, hence getting used to the kind of problem analysis which is
necessary for algebra applications. Passing from the spreadsheet
style of resolution to the algebraic one requires learning to synthesise
the partial relationships presented in the spreadsheet into one or more
equations describing the problem. The plain use of a spreadsheet,
without teacher's help, will hardly lead the student to acquire this
synthesis capability. At the same time, the possibility to find a
solution simply by numerical attempts (which are easy to perform by
computer) can discourage the student from making an intellectual
effort toward synthesis, hence missing a fundamental tool for the
development of mature cognitive capabilities.
Fig. 2. Formulas for problem 1
Variables and Unknowns
Another component which is absent from the spreadsheet solution is
the unknown X. Since formulae computed by spreadsheets are not
relations but functions, the involved cell names at most play the role
of functional variables rather than algebraic unknowns. However, the
functional variable is the result of an abstraction process, which is
the capability to figure out an object beyond the possible values that
can be substituted for it. As in the case of synthesis, abstraction
capability will hardly be achieved by students with the only help of a
spreadsheet, without support of a teacher who can suggest it by
pointing out the analogies and differences between rows obtained
from one another with a copy instruction (see fig. 2).
Proving results
Let us now consider another problem, slightly more complex than
the previous one: The theatre of a country town has 100 seats,
divided into front section and rear section. The price of front seats is
8$, that of rear seats is 6$. When all seats are sold, the total income
is 650$. How many front seats and rear seats are there in the
theatre?
The algebraic resolution is made up of two equations in two
variables, X+Y=100 and 8X+6Y=650, which give in a simple way
the solution of the problem as X=25, Y=75.
If the spreadsheet solution is designed without implicitly solving the
first of the above equations with respect to one variable, we obtain a
table which is clearly too cumbersome, but bright students will of
course almost immediately notice that for each X value it is
necessary to try only one Y value, otherwise their sum may not be
100 (see fig. 3). However, being unable to make formal proofs, it
will be difficult to argue that the solution found is the only one,
while the algebraic solution gives this certitude. On the other hand,
tackling situations of this kind with a spreadsheet, the teacher can
stimulate students to reason about the range and relationships of
possible solutions to a problem.
Introductory study of functions
An interesting aspect of spreadsheet resolutions, as appears from the
above example and considerations, is that these tables are potentially
dynamic, that is, columns and rows can be added (e.g. row 12 in fig.
3) or deleted in a much easier way than with paper-and-pencil tables,
hence making it possible to solve more problems with a same table,
or to seek a more accurate solution of a problem. Moreover, the
facility of building and modifying tables leads more easily to their
use as a tool to make and to test conjectures about a function's trends.
For these reasons, the spreadsheet can be a good introductory tool for
the study of functions.
Fig. 3. Numeric resolution and formulas for problem 2
Generalisation of problems by means of parameters
The spreadsheet can be useful to introduce the concept of
generalisation of a problem and to learn to distinguish between
variables and parameters. For instance, the problem considered
above can be used, with the same structure but with different data, to
describe the case of different theatres, and it is important that
students learn to recognise its unchanged structure. The presence of
values that can be considered as parameters can be emphasised in the
spreadsheet by writing them separated, before the description of the
problem (see cells A2, B2, C2 in Fig. 4). These cells are then
referred by using the "absolute notation" (e;g. $A$2), which shows
that the cell name is not to be changed during the copy-and-paste
operations. The use of relative and absolute references emphasises
the difference between parameters and variables.
Fig. 4. Emphasising parameters in problem 2
The relationship between parameters and variables can be stressed by
considering the dual of a problem. For example, in relation with the
previous problem, let us consider the following one: In occasion of a
special show, more expensive than the usual ones, the management
of the previously considered theatre needs to increase the price of all
seats. The prices ratio must remain 4 to 3, and the total income must
be $975. At what price must front and rear seats be sold? In order to
resolve this problem using a spreadsheet, it is sufficient to exchange
the roles of variables and parameters in the previous table, as shown
in fig. 5. Though the formulas in column B are slightly changed,
those in column C are exactly the same.
Fig. 5. Numeric resolution and formulas for problem 3.
Problems with more than one solution
A different case is that of false generalisations. Let us consider for
instance this problem:
The theatre of a country town has 100 seats, divided into first,
second and third seats. First seats cost 9$, seconds ones cost 7$ and
third ones cost 5$. When all seats are sold, the income is 700$. How
many first, second, and third seats are there in the theatre?
Seeking a solution by using the spreadsheet, it is evident that it is
more difficult to chose suitable values by trials, but it is not evident
where the difficulty comes from. It is clear that one of the three
values is determined by the other two in order to make their sum
100, but this does not help to find a solution, and certainly does not
make clear that in this case, unlike in the previous one, there are
many possible solutions (see fig. 6). An algebraic treatment of this
problem, on the contrary, shows that the two problems are
structurally different, since in this case we have three variables and
only two equations, hence the problem has an infinite number of
solutions in R, all characterised by the relations X=Z, and Y=1002X, which allows us to chose a finite number of reasonable solutions
in N. This is certainly not easy to see if the problem is tackled only
by spreadsheet, without any algebraic consideration.
Fig. 6. Numeric resolution for problem 4
Conclusions
This a-priori analysis pointed out that the spreadsheet can be useful
to introduce algebra, since it leads to recognising the elements
involved in a problem and to expressing part of the relationships
among them. On this basis, numerical solutions of simple problems
can be easily found. The limitations of this environment as concerns
teaching/learning algebra are due to several factors:
• spreadsheets deal essentially with numbers, or addresses of
numbers, and functions;
• algebraic variables and relations can not be directly handled in a
spreadsheet; only assignments are made;
• spreadsheets operate from "knowns" to "unknowns", which is the
opposite of what characterises the algebraic thinking.
Moreover, spreadsheets are unable to formally manipulate relations,
hence useless to learn to construct formal demonstrations, which are
one of the main applications of algebra.
However, using a spreadsheet, which by itself would lead students
to solve problems only by trials, under the wise guidance of a teacher
can lead:
• to activate a modelling process for problem resolution;
• to understand what means to solve an equation, even before
knowing what an equation is (that is, to find a value such that an
expression is valid);
• to reason on the constraints of a problem in order to decrease the
number of trials necessary to find a solution (from casual to
focused trials);
• to introduce the concept of approximate calculus.
Further steps toward a real learning of basic algebra can be made
through a reflection, strongly guided by the teacher, on the resolution
model implemented by means of the spreadsheet. In fact, the
teacher's role appears essential:
• to guide her students to abstract the concept of algebraic
variable, not present in the spreadsheet, by remarking the
analogies of formulas repeated in different rows;
• to show to her students how to synthesise equations describing
the problem, based on the formulas used in the spreadsheet and
on direct numerical checks on them;
• to make her students aware of the fundamental diversity of the
operator "=" in the spreadsheet and in algebra, that is,
assignment vs. relation (without this distinction, using a
spreadsheet can even be misleading);
• to introduce problem generalisation by differentiating parameters
and variables;
• to reinforce the student's algebraic competence by comparing, as
a metacognitive activity, different problem solving
methodologies, such as arithmetic, algebra and spreadsheet.
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