Creative Education, 2010, 2, 115-120
doi:10.4236/ce.2010.12017 Published Online September 2010 (
Copyright © 2010 SciRes. CE
Using an Interactive Computer System to Support
the Task of Building the Notions of Ratio and
Elena Fabiola Ruiz Ledesma
Escuela Superior de Cómputo del Instituto Politécnico Nacional, Departamento de Posgrado, México.
Received February 27th, 2010; revised July 18th, 2010; accepted July 25th, 2010.
This article describes the design and general outcome of app lying a computer system tha t includes interactive a ctivities
to the topics of ratio and proportion. The work was undertaken with Mexican students attending primary school grade
six (11 year olds). In designing the activities, our work was based on the studies of researchers who have focused on
such topics, as well as on the work of researchers in the field of computer scien ces and education techno logy. Support
of the activities designed is found in psycho-pedagogy, in knowledge of mathematics and in the fields of computer sci-
ences and education technology.
Keywords: Ratio, Proportion, Interactive Activities
1. Introduction
There is little doubt of just how important it is that stu-
dents attending b asic education classes develop cogn itive
structures in the area of mathematics, structures that pro-
vide them with the basis needed to successfully meet the
academic demands of the next school system level.
Hence development of software for basic education in
mathematics becomes just as significant in the current
Mexican context because incursion of computers into
that educational level raises the need to analyze which
pedagogical strategies recommended by Mexican re-
searchers to teach the concepts of ratio and proportion
can be incorporated into an educational software system,
and precisely how that should be done.
Consequently the authors of this article have focused
their attention on supporting the task of building the no-
tions of ratio and proportion. And in doing so, they have
developed a computer system that uses technology to
carry out interactive activities.
In this article, the authors introduce a teaching pro-
posal developed in a computer system with interactive
activities that make us e of technology. The design of the
proposal activities was based on a pr ior study undertak en
by Ruiz, [1] with students who course six grade of pri-
mary school and that deals with the topics of ratio and
proportion. These topics were selected in view of the fact
that their instruction is begun in primary school and that
they serve as the basis for subsequent concepts, such as
those of direct proportional va ri at i on or li near function.
2. Portrayal of the Problem
To identify whether the ratio and proportion teaching
activities designed through a computer system enable the
11 year-old student to build notions or ratio and propor-
tion, developing said student’s qualitative proportional
thought and aiding him/her to recover the sense of his
qualitative proportional thought.
Qualitative proportional thought is supported by lin-
guistic recognition, creating comparison categories such
as large or small. Qualitative thought also in clud es things
intuitive that are based on experience and that are em-
pirical, and things perceptual that are supported by the
Qualitative proportional thought refers to activities
that enable students to count measure and employ quan-
tities in procedures.
3. Theoretical Basis
3.1. On Ratio and Proportion
According to Piaget [2] one can see in 11 and 12 year old
subjects the presence of the notion of proportion in dif-
ferent areas, notions such as: spatial proportions (like
Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion
Copyright © 2010 SciRes. CE
figures), the relation between the weights and length of
arms on a scale, probabilities, etc. [3], also states, based
on his experiments, that children acquire their qualita-
tive identity before their quantitative conservation, and
further distinguishes between qualitative comparisons
and true quantification.
In fact for Piaget the notion of proportion always be-
gins qualitatively and logically prior to being structured
from a quantitative standpoint. He stresses that in order
for a student to develop his / her qualitativ e proportional
thought that student must necessarily begin with the no-
tions of enla rgement and reduction, followin g the idea of
a photocopier or scale drawing, assuming that at a very
early age the student is able to recognize what is propor-
According to Piaget and Inhelder [3] after the student
develops his / her perceptual ability, an ordering takes
place by way of comparisons, which can be seen when
students use phrases the likes of “larger than…” and
“smaller than…” and which are known as verba l catego-
ries. In this sense Piaget says that during the transition
from things qualitative to things quantitative, the idea of
order appears without a quantity having yet emerged, an
event that Piaget calls intensive quantifications.
Subsequently when students use measurements to
make comparisons they first confront parts of the object
and superimpose one figure on top of another and after-
ward use a measuring instrument, be that a conventional
instrument or not. Freudenthal [4] states that compari-
sons make it possible to measure and measurement is
shown by way of two modes: direct and indirect. The
direct mode of comparing is when an object is superim-
posed onto another object, while the indirect mode is
when there are two objects (A and B) and a third element
to compare (C).
With respect to the emphasis that should be placed on
early education of ratio and proportion, Streefland [5-7],
states that the point of departure should be qualitative
levels of ratio and proportion recognition and that use
should be made of didactic resources that foster devel-
opment of perceptual patterns in support of the corre-
sponding quantificatio n pr ocesses.
The didactics of mathematics are referred to as an
essential activity in teaching ratio and proportion, as is
the importance of didactic tools developed by the de-
signer. Freudenthal’s Didactic Phenomenology [4] is
particularly mentioned, together with other background
considered in order to attain a realistic building of ma-
Freudenthal indicates that comprehension of ratio can
be guided and deepened by way of visualizations. He
moreover states that such visualizatio ns can be illustrated
by way of detailed constructions in which the drawings
are differentiated and in which the drawings depict
which points of the original and image actua lly coincide.
An example of the foregoing is in two contig uous figures
one of which is an enlargement or reduction of the othe r,
and in which the same linear ratio can be established in
each segment of the figure. Freudenthal suggests that
when working with the ratio of longitud es, flat figures is
used as a means of representation because of their global
expressiveness, in the sense that a student’s qualitative
and quantitative comprehension is facilitated by visual
Ruiz and Valdemoros [8] found that the students she
worked with presented differen t difficulties, specified as:
a) The students’ qualitative though t dealing with propor-
tionality has not been explo ited to the utmost, which was
observed when they demonstrated that they were cen-
tered on one of the dimensions of the figures they had
been asked to reduce or expand; b) In some students,
things qualitative are barely raised as an introduction to
things quantitative, given that in the linguistic categories
detected among them, one found the following: “it is
larger than…”, “it is smaller than…”, which reflects a
certain understanding of proportion; yet among these
same students one did not find other categories through
which they showed a greater understanding of the idea of
proportion; c) They appeared confused when establishing
relations between quantities, which is why is became
necessary to emphasize that in order to reach the notion
of ratio.
3.2. Technological Elements
Lim [9], states that technology makes it possible for
teachers to be more flexible thus enabling them to ad-
dress the different needs of students with varying levels
of capabilities – all of whom may be sharing one single
classroom- by using software that can be adapted to the
teaching and to the particular conditions of each student
or group. Technology makes it possible for teachers to
divide their groups of students into teams and to work
with each one at their own pace. What is more, technolog y
can make it possible for the students to determine the
pace at which they feel comfortable working.
Some research has been undertaken for the purpose of
finding out if usage of a Web-based environment bene-
fits learning. The study carried out by Galbraith and
Haines [10] shows that students who use a computer in
their daily mathematics learning enjoy mathematics.
They like the flexibility provided by the computer and
spend a great deal of time at the computer to complete a
task, and enjoy trying out new ideas. The researchers
also concluded that Web-based applications increase
levels of confidence, motivation and interaction.
Nguyen and Kulm [11], Combs [12], Gourash [13] and
Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion
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Engelbrecht and Harding [14,15], point out that usage of
computers in educational terms enables students to find
the meaning of what they are doing in that their ability to
discover is developed and they are able to delve deeper.
Working with the points mentioned by the researcher
in the previous paragraphs, Table 1 depicts a specifica-
tion of the indicators that refer to ratio and proportion, as
well as to the didactic actions associated to those indica-
4. Methodology
Under this heading the authors of this article have in-
cluded the subjects with whom work was undertak en, the
elements of the computer system used to carry out didactic
activities and the didactic activities themselves.
The research was carried out with a sample of 29 stu-
dents all in the same grade six class of a primary school
in Mexico City, to whom the activities designed were
Sequence of activities and their presentation in the
computer system
Taken into account in the task of designing the se-
quence of activities are the didactic actions specified in
the theoretical framework, the computer activities, as
well as the theoretical elements entailed in the concepts
of ratio and proportion.
Activity 1. Choose the reduced or enlarged figure
of the figure provided, by way of visualization.
This activity is based on ideas of “reduction and
enlargement supported by models of the scale and
photocopier drawings-type experiment, using things
perceptual and observation.
This was used as the point of departure in view of that
indicated by Piaget, [12], Streefland, [5-7] and Ruiz y
Valdemoros [8], regarding the point that early instruction
on ratio and proportion needs to begin with qualitative
levels of that learning, which is why activities that do not
require use of quantities for solution of the activities are
first used.
Table 1. Objectives, indicators and didactic actions of the concepts of ratio and proportion.
Concepts Objectives Indicator Didactic Actions
Intuitively establish ratios Compare Superimpose figures
Use verbal categories such as “one side fits twice in the other
side” or “one side is one third of the other side”
Relation between
two magnitudes
through a quo-
tient Explicitly establish ratios Express the ratio in the
form of a fraction
Count the sides of a square (in a grid)
Use a table
Count the sides of a square and wri te out the r a ti o as a frac tion .
Qualitative proportional
-Use linguistic
Choose reduced or expanded fig ures.
Enlarge and reduce figures.
Use verbal categories, such as “larger than” or “smaller than”
Transition from qualitative
to quantitative propor-
tional thought
-Measure indirectly Superimpose figures
Count sides in the squares of a grid
Proportion: Re-
lation of ratios or
the equivalence
of two or more
ratios Quantitative proportional
-Measure directly
-Use the rule of three or
of the excluded third
Measure using a conventional instrument
Use the table
Carry out number operations
Table 2. Contains the relation existing between Table 1
activities and the actions to be undertaken on the computer.
Didactic Actions Actions on the Computer
Superimpose figures Drag the mouse
Use verbal categories Use voice
Count the sides of a square
(in a grid) Use a pencil, like in Paint
Choose reduced or enlarged
figures Click on the option
Expand or reduce figures Use a pencil, like in Paint
Measure using a conventional
instrument Use ruler
Use the table Table to the filled in by the
Carry out number opera ti ons Calculator
Consideration has furthermore been given to Freuden-
thal’s points that deal with guiding and deepening com-
prehension of ratio through visualizations.
By the same token the figures designed in the com-
puter system relate to the knowledge possessed by 11
year old children, to whom the figures are aimed. Exam-
ples of the foregoing could be a boat, a bus, a star, a dog,
albeit all drawn in straight line seg ments.
How the concepts appear in the computer system
Shown are four similar figures, all of which have
small differences. One of the four figures is also depicted
in a double or triple linear amplification or reduced by
half or to a third of the linear size. The user is asked to
choose among the four figures and find the figure that is
a reduced version of the original. (See Figure 1).
Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion
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Figure 1. Source: Created by the authors.
After having chosen one of the figures, the choice is
then analyzed and an answer is immediately provided
bearing the result of the analysis. The user is asked if he /
she wants to try again, the answer is received and, de-
pending on that answe r, the exer cise is either restarted or
work is continued with the next exercise.
Activity 2. Choose the figure that is a reduction of
amplification of the figure provided, by way of com-
Case 1 Superimpose one figure over another
Superimposition of one figure on top of another en-
ables subjects to recognize relations of similarity among
the figures in intuitive terms.
The action of comparing figures is the beginning of
measurement, yet without using a conventional instru-
ment since it is achieved by superimposing the figures
just as specified by Freudenthal, 1983.
How the activ i t y appears in the comp ut er system?
The student has the option of using his / her mouse to
drag any of the four figures in order to place the figure
on top of the original and review the result, by way of
visualization, to see if the figure is an enlargement or
reduction on all sides by the same amount. See Figure 2.
Case 2 Use of a grid
The transition from qualitative to quantitative propor-
tional thought is achieved by using the grid [1] Counting
is used and the measurement unit is one side of a square
in the grid in which the figures are portra yed.
The result of the process of counting the sides of the
figures is used in order to establish quotient relations
with the results obtained, th at is to say ratios. This is also
done for the purpose of establishing relations of equiva-
lence between t wo rat i os or proportions.
How the activity appears in the computer s ystem
The student is presented with a figure on a grid, as
well as another empty grid in which he / she can draw
and use the grid itself as a means of support. The student
Figure 2. Source: Created by the authors.
is asked to draw the figure at double or half or one third,
etc., of its original size (See Figure 2). The drawing in
the first grid is a means of support, used to compare with
the drawing done by the student. The student is told
whether his / her answer is correct and asked if he / she
wants to repeat the exercise, and the student decides if he
/ she wants to repeat the exercise or go on to the next
Activity 3. Use of the table
The table was used as a means of representation in or-
der to determine internal and external ratios Freudenthal
[4]. The students work with proportional variation prob-
lems and obtainment of the quantities is not only
achieved through use of the operator, but also by estab-
lishing relations among ratios.
Finally, work was done on equivalence relations as a
proportionality relation.
Outcomes of the didactic activities
Activity 1. Choose the figure using visualization
Eleven of 29 students used observation to choose the
reduced figure. Whereas the remaining 18 decided to
compare the figures by superimposing one figure over
the other s. They used the mo use to drag the figu re to see
if it was, from the side or width, double, half or one third
of the original. All of this was commented upon during
the session; and furthermore coincides with Freudenthal
[4]. If the figure in question was a circumference, the
students compared the radii or the diameters. Conse-
quently all of the students were ab le to determine that the
answer was the reduced figure, and each student decided
for him / herself whether to drag the figure and superim-
pose it over the other figure in order to compare them.
This student reaction is similar to comments made by
Combs [12].
Activity 2. Grid
Of the entire group of 29 students, 23 were able to
draw correctly on the grid figures that were similar to
Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion
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those presented to them. The remaining 6 students made
two to three attempts before they were successful with
the task. The computer system was extremely useful in
that it presents the student with the option of working
with different figures, thus making for work that is not
mechanical. What is more, students are given the oppor-
tunity to discover what is taking place.
The work reached the level of recognition of ratios as
a comparison by quotient of two quantities. The group
worked on notation of ratios in the form of a / b fractions,
in which b is a different from 0.
Activity 3. Use of the table
Once again of the entire group of 29, 20 students were
able to fill in the table correctly. They accomplished this
by deciphering the unit value, in the ev ent that it was not
provided, wh ile others established ratios by reading them
from the table and writing them down in the form of
fractions. All students used the calculator, first deter-
mining the operations that had to be undertaken; for in-
stance, in Figure 3, reference is made to the case of Luis.
When faced with the activity, this is what Luis did:
In the case of Luis, one can see that he establishes rela-
tions between two variables, the milk and the chocolate
bars. In Freudenthal terms, these are external ratios.
Manuel is another case: What Manual expressed when
solving the activity can be seen in Figure 4.
Manuel was asked to provide a conclu sion, as follows:
The conclusion youre asking me for is that well like, to get
the [number of] chocolate bars you have to use the 6 times table. So,
first you need 6 bars for 3 liters of milk, then 12 bars for 6 liters of
milk, then 18 bars for 9 liters. So if I compare liters to chocolate
bars I can see that its double the number of liters in chocolate
There is indeed food for thought in Manuel’s work.
Before asking him to provide a conclusion, Manuel had
already established relations among the data in one of the
columns, the liters of milk column. Freudenthal refers to
this as internal ratios.
Usage of the table enabled the students to establish
relations between magnitudes of figures, to the point of
establishing both internal and external ratios (as per th eir
definition by Freudenthal). Lastly the students were in-
I saw that 3 fits 2 times into 6, so for
3 liters of milk you need 6 bars of choco-
late. In other words, you need double the
amount of chocolate bars, so one liter
needs two bars, two liters four bars, three
liters six bars.
Ill just use the calculator to multiply
by two to fill in the data I’m asked f o r
Figure 3. Response provided by Luis in the activity of filling in the table.
I saw how the quantities in the choc-
olate bars column changed, because they
gave us more numbers in that column. I
saw that it began with 6, then there was a
blank that I had to fill in, then came the
18 and right after that the 24. I used the
calculator to divide 18 by 6, and it gave
me a result of 3; then I divided 24 by 6
and it gave me 4. So I took the 6 and
multiplied it by 2 and got 12, and thats
the value of the second blank, and the last
blank in the column is the result I got
from 6 times 5, which is 30.”
Figure 4. Manuel fills in the table, first filling in the liters of milk column.
Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion
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deed able to use the different representation registers –
drawing, table and things numerical- when they were
solving ratio and proportion problems. One is therefore
led to point out that work with th ese interactive activities
enabled the students to build the concepts of ratio and
proportion, investing them with meaning and sense, ra-
ther than just using the algorithm to work on the topics.
This is indeed similar to the finding s obtained in the stu-
dies by Galbraith, P. and Haines, C. [10], Nguyen and
Kulm [11], Combs [12] and (Engelbrecht and Harding
5. Conclusions
According to the findings obtained we are in a position
to state that the activities de signed to support students in
the task of consolidating the concepts of ratio and pro-
portion were adequate given that the students demon-
strated a great deal of interest when working with the
computer system and worked independently to solve the
problems. Their teacher did not have to instruct the stu-
dents to compare one figure with another in order to de-
termine which one was the reduction, rather the students
developed that ability than ks to the activities propo sed in
the system. The foregoing leads one to infer that the stu-
dents were able to develop both their qualitative and
quantitative proportional thought; they showed a good
deal of freedom in dragging the mouse, using the grid
and filling in the table, activities for which there was no
need for their teacher to give them instructions. Working
in this manner moreover enabled the students to develop
visually and perceptually, in other words they were able
to further develop asp ects that constitute their qualitative
proportion a l thought.
Use of the table made it possible for the students to
establish relations among figure magnitudes, establishing
internal and external ratios (as defined by Freudenthal).
Finally the students were able to use the different regis-
ters of representation (the drawing, the table and things
numerical) when solving ratio and proportion problems,
which enab les the authors of this article to point o ut that
working with interactive activities made it possible for
the students to build th e concepts of ratio and prop ortion,
and imbue those concepts with meaning and sense rather
than simply using the algorithm to work on the topics.
This is quite similar to the findings obtained in the stud-
ies undertaken by Galbraith, P. and Haines, C., Nguyen
and Kulm, (Combs, and (Engelbrecht and Harding, The
computer system made it possible for the students using
it to develop skills that were not developed by all of the
other students in their group using pencil and paper and
the blackboard.
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