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Verónica Vargas
José Guzmán
Cinvestav-IPN y CIDEM, México
Cinvestav-IPN, México
In this article we report the detailed analysis of the procedures employed by a group
of high school students in approaching a verbal algebraic problem involving velocity.
This is a part of a sequence of 17 problems of rate that were implemented in the
classroom utilizing paper and pencil environment, with the end of promoting the
evolution of algebraic thought in high school students. The results demonstrate that
these students, in spite of having previous assisted in algebra courses, continue to use
arithmetical type procedures in their solution processes, and from this we suggest
that it is necessary to place more emphasis on the multiplicative relations in the
practice of teaching.
The work of translating between natural language and algebraic language is a
challenge for pre-algebraic students (Malara, 1999). Even when they are able to
identify the relationships between the elements of a given problem, there exists the
difficulty of expressing themselves in an algebraic code. In other words, the students
do not know how to relate that code with the semantics of natural language (Bazzini,
In this article we report in detail on the procedures used by a group of high school
students in approaching a verbal algebraic problem (we will say single problem)
involving velocity. The problem analyzed was part of a larger investigation working
with a sequence of 17 problems of rate that were implemented in the classroom for
the solution of problems, with the proposition of promoting the evolution of algebraic
thought in the high school students. An addition, we propose to answer the
questions: How do high school students perform in the confrontation of problems
involving velocity? What are their procedures for solution? What type of thought
processes are characterized by the students in the use of those procedures?
The analysis of the procedures employed in the solution of the problem was based on
research carried out by Bednarz & Janvier (1994, 1996) and Vergnaud (1991).
Literature Review
Bednarz & Janvier (1994, 1996) proposed a theoretical tool (grille d’analyse) that
permits the classification of word problems utilized in the teaching and learning of
arithmetic and algebra. One of the three classes of problems that were identified
were those of rate which involved a relationship of comparison between nonhomogenous magnitudes. Problem 1 is an example of this type; we can observe two
2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the
internacional Group for the Psychology of Mathematics Education, Vol. 5, pp. 313-320. Prague: PME. 5 - 313
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non-homogenous magnitudes (distance and time) related through a rate (velocity).
The symbolism created by these authors to sustain their theory is described in
Guzmán, Bednarz & Hitt (2003, p. 11).
Bednarz & Janvier (1996) defined the characteristics that determine the difference
between an arithmetical problem and an algebraic one. The algebraic, or
disconnected, problems were characterized by the difficulty presented in establishing
direct bridges between the known quantities. These must be operated with various
unknown quantities at the same time, making it necessary to establish equations in
order to solve them (Figure 1). On the other hand, it is possible to solve a connected
problem using the given data by “progressive negation of cases”. That is to say, in
these problems it is possible to construct bridges between the known states and
relationships to obtain the unknown states and relationships. In general, in this way
the arithmetical procedures are organized according to the processing of known
quantities. The students create unions between these and finalize the operations; the
unknown quantities are left until the end of the process.
From the above, we can derive the characteristics of algebraic thought or a typical
profile of a solver.
Knowledge of patterns (regularities).
The reading of the problem
(knowledge of the quantities involved in the problem as well as the
relationship between them) and the resources which occur to the student in
order to understand it are the essential elements of this phase.
Establishment of conjectures. The phase of formation of a mental
representation that interprets the information given in the problem and
transforms it into objects with its associated properties, organizing the
relationships between these objects and representing the relationships
through equations.
Symbolic manipulation. The phase of solving equations which requires the
knowledge of manipulating them.
The comprehension and management of the basic elemental operations are essential
in algebraic thought. The multiplicative relationships (multiplication or division),
and in particular the relationships among four quantities (Vergnaud, 1991) underlie
the structure of many arithmetical and algebraic problems. These relationships occur
among four quantities, two quantities of a certain type (for example kilometers), and
the others of another type (for example, minutes). Vergnaud (1991) proposed
schemes of tables of correspondence in order to analyze the complexity of problems
that involve relationships among four quantities and made an analysis of the type of
operations that the student could do. This author placed emphasis on two types of
operations, scalar and functional, and suggested that for the students, whose thought
is arithmetic, the functional operations were of higher complexity, given that these
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involve not only the notion of the numerical relationship but of the knowledge of
In this article we discuss the procedures followed by a group of students toward the
following problem involving velocity.
Problem 1. The highway that connects
cities A and B, from A towards B, goes
up- and then downhill. A bicyclist,
30 km/h
whose average velocity is 10 km/h on
the rise and 30 km/h going downhill,
takes an hour and 30 minutes to get
from A to B and two hours and 30
2 h y 30 min
1 h y 30 min
minutes to bet from B to A. Calculate Figure 1. Scheme of the Problem 1. This
the distance to each of the cities from type of scheme was explained by Guzmán
the highest point of the route. (Bednarz
et al. (2003).
et al., 2003, p. 13.)
Figure 1 shows the scheme of the structure of the problem, taking as reference the
theoretical tool of Bednarz & Janvier (1996). The rate is velocity (km/h). This rate
relates the two non-homogenous quantities of distance and time.
The problem was implemented in a school which is part of a technical public high
school in the state of Michoacan, Mexico. The group was comprised of 45 students
who were in their first semester. They were selected for being at the end of their
semester of algebra, which is in the first semester of high school. Because of this,
their basic knowledge in this discipline corresponded to their algebra course in
secondary and high school.
In a work session in the classroom the group in the study, made up of teams of three
students, was given Problem 1. They were given approximately 30 minutes to solve
the problem, followed by a group discussion of the solution procedures employed.
The session ended with a general presentation by the group of the results obtained
and the discussion of the procedures of solution. This session of the work in the
classroom was both video taped and audio taped.
Discussion of the solution of Problem 1. This problem is not connected or algebraic,
in Bednarz & Janvier’s (1996) terms. The possible diagrams (the meaning of the
diagram is used in a different manner that that of the scheme by Bednarz & Janvier,
1996) that were represented are the following.
v s = 10 km / h
vb = 30 km / h
vb = 30 km / h
v s = 10 km / h
2.5 h
1.5 h
Figure 2. Diagrams representative of Problem 1. The data of an hour and 30 minutes
is represented as 1.5 h.
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This problem can be solved in an algebraic manner by writing and solving a system
⎧t + t 2 = 1.5
which involve the relationship v = ,
⎩t 3 + t 4 = 2.5
of equation such as the following ⎨ 1
where ti (i = 1, 2, 3, 4) represents the times required by the bicyclist for both
segments of the trip. The system of preceding equations could be transformed to:
⎧ d1 d 2
⎪⎪10 + 30 = 1.5
⎪ d 2 + d1 = 2.5
⎪⎩ 10 30
Where d1 represents the distance from A to the highest point of the
highway and d 2 represents the distance to the highest point in the highway. When
⎧ d 1 = 7 .5
in kilometers.
⎩ 2
solving this system of equations the solution is ⎨
The solution to this problem cannot be determined through arithmetic procedures as
the problem is disconnected. However, if a solver makes connections by trial and
error it is possible to come to a solution. One possible form to make the connection
is the following. Propose a value for t1 (Figure 3) and employ this to calculate the
values of t 2 (through the relationship of t1 + t 2 = 1.5 ) and of d1 (through the
); then calculate d 2 , t 4 and t 3 . These last values serve to verify
the relationship of t 3 + t 4 = 2.5 .
relationship of v =
30 km/h
1 h y 30 min
2 h y 30 min
Figure 3. Scheme of the structure of Problem 1, after making connections.
Numerical example, utilized in the scheme in Figure 3. Given that the time taken for
the cyclist in the run from A to B is one hour and 30 minutes, it can be supposed that
one hour is the time taken to go from A to the highest point on the highway (P) and
0.5 hours is the time taken to go from the highest point (P) to B (Figure 4a).
10 km / h
30 km / h
0.5 h
30 km / h
10 km
10 km / h
15 km
Figure 4. Diagram representative of Problem 1. In (a) numerical values have been
assigned to the times of the trip from A to P and from P to B and in (b) is shown the
values obtained for the distances asked.
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With these values and considering the relationship of v =
, the distances from A to
P and from P to B are calculated as 10 km and 15 km, respectively (Figure 4b), which
can be verified by using them to calculate the times taken by the cyclist in the return
to B from A. In this case, the times taken by the cyclist from B to P and from P to A
are 0.33 hours and 1.5 hours, respectively, whose sum is not the hoped for one of 2.5
hours. Because of this, the process must be reinitiated; students have to suppose
other values for the times taken by the cyclist to make the distances from A to P and
from P to B in order to obtain 2.5 hours.
We hoped to observe these two solution process, the algebraic and the arithmetical, in
the procedures of solution used by the students, primarily the algebraic.
Results and discussion
Half of the group of students attempted to solve the problem through the formulation
of a system of equations but none successfully established in a correct manner the
equations because it was not clear to them how to relate the involved quantities in the
problem; they did not encounter an adequate representation of the variables in
The other groups of students attempted to solve the problem through arithmetic. The
majority of them did not succeed. Only one group successfully solved the problem in
the correct manner. The first phase of the solution process of this group consisted in
the knowledge of the known and unknown quantities of the problem (Figure 5).
Represented therein is part of the information given in the problem. Following this
they converted the hour and a half used by the cyclist to go from A to B into 90
The second phase consisted in making conjecture. The team of students set out the
need to know the times that were taken by the cyclist to go the distances from A to
the highest point of the highway, and from this point to B; because of this, they
divided 90 minutes by two and considered that the cyclist took 45 minutes to go from
both sections. This permitted them to connect (Figure 3) the problem. Here is the
explication that the speaker for the group gave to the rest of the class with respect to
the solution process that had been utilized.
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What group wishes to make their presentation?
We do. (A student from the group went to the board with one of the
papers on which were written the procedures –Figure 5– and began to
draw a diagram without saying anything. The others waited for the
explanation while watching and making commentaries. Some were
following what the student was writing. There was a lot of noise in the
We must pay attention to see how they solved the problem.
Here what we did was to divide the hour and a half (the student began to
speak, but as the noise continued in the classroom the explanation could
not be heard, and he was asked to start over.)
(Interrupting the speaker) We cannot hear you at the back.
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(Raising his voice) We divided the hour and a half that it took from A to B
(pointing with his finger to points A and B of the diagram) and this gave
us 45 minutes to go uphill and 45 minutes to go downhill (pointed again
towards the diagram of the rise and fall of the highway and continued the
explication). Then we saw that so many kilometers were travelled in 45
minutes on the rise and this gave us 7.5 kilometers, and going downhill
there were 22.5 kilometers and then the reverse the 22.5 kilometers gave
us 2.15 hours on the uphill part…
Hey (an expression trying to confirm what was said by his classmate), and
going downhill (this student, apparently, following the explication of his
and the 7.5 kilometers (uphill) give 15 minutes, and thus it was as we
obtained it…um, 45 and 45 are an hour and a half from A to B. The 15
others the two hours and 15 minutes gave us the two hours and a half.
Oh, ah (There were many doubts as much on the part of the students as on
that of the researcher. However, they permitted Student 1 to finish his
explanation. Also, this student did not pay attention to the questions of his
classmates because he was concentrating on explaining in a summary
manner the solution process).
In other words, there are two (made a mistake and corrected it), six
minutes taken per kilometer uphill […] (The student continued talking but
the other students started to repeat what he was saying, trying to follow
his explanations in order to understand them and stopped listening to
(Interrupting the student’s speech) Where did the six minutes […] per
kilometer come from?
We divided the 60 minutes of an hour by the 10 kilometers of the uphill
part and…
60 by 10 kilometers, 60 minutes by the 10 kilometers gives 6 minutes (he
had difficulty with the dimensions. In reality, the only thing that was
controlled was the operations with the numerical quantities. The
relationships of the dimensions were not being made, and because of this
he at times mentioned minutes and at times did not, and the same was
done with the kilometers.)
Figure 5. Solution procedures of Problem 1 utilized by a group of students.
Figure 5 illustrates the diagram in which the previous explanation was based, but the
diagram utilized for this was a reinterpretation of the registered process done with
pencil and paper to solve Problem 1.
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Continuing with the discussion of the procedures, the team of students employed the
relation of equivalence of one hour and 60 minutes to find what distance the cyclist
covered in a certain quantity of minutes. The reasoning can be explained using
correspondence table of Vergnaud (1991) between kilometers and minutes (Figure 6).
÷ 10
× 10
÷ 10
Figure 6. Table of Correspondence between two types of quantities: kilometers and
The team of students parted from the link of correspondence between the 10
kilometers and the 60 minutes and looked for a unitary value. But they did not
decide to look for so many kilometers for each minute taken by the cyclist but for so
many minutes taken by the cyclist to cover 1 kilometer. Here intervened the notion
of division, 60 was divided by 10, in order to show the vertical relationship from high
to low (Figure 6) in the left column. The operator ÷10 is a scale operator that
reproduces in the right column what occurs in the left, which expresses the change
from 10 km to 1 km and is the inverse operator of the operator x10, which permits the
change from 1 km to 10 km. The students did not think in terms of horizontal
operators (functions) that would permit the change from kilometers to minutes but in
terms of vertical operators (scalar). According to Vergnaud (1991), to finalize
horizontal operations requires a level of complexity higher than to finalize vertical
operations and is, additionally, the reason for the difficulties encountered in helping
the student understand the notion of function.
Knowing that the cyclist took 6 minutes to cover 1 kilometer when going uphill and 2
minutes to cover 1 kilometer when going downhill permitted the students to calculate
how much time, in minutes, it took the cyclist on the up- or downhill distances, which
helped to find the distance that the cyclist covered in 45 minutes of both up- and
downhill travel, 7.5 km and 22.5 km respectively. The team of students finalized the
operations with these distances and found that both complied with the time taken by
the cyclist from B to A.
The process of solution employed by this team of students is arithmetic, as they
connected the problem, and afterwards finalized the operations to find the unknown
distances. Given that they connected the problem, it was not necessary to establish
equations. Their first supposition was apt in the sense that they elected the correct
values for the times, but if they had not done this they would have continued
assigning values to the times of travel (and through the management of the involved
relationships) until confirming that, effectively, the distances obtained were those
they were looking for.
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The characteristics of their thought are of the arithmetical type in spite of the fact that
in the first phase (reading of the problem) the solution process the group was able to
recognize the quantities involved as well the relationship between them, in the second
phase of the process (establishment of conjectures) they did not interpret this
information in algebraic objects and continued manipulating numeric quantities;
because of this they did not write equations. Also, they used scalar operations (or
vertical in the sense of Vergnaud, 1991), which denotes a fault of conceptual
development of the multiplicative relationships.
However, given that this team was able to recognize the structure of the problem and
the relationships involved we can say that this team was closer to algebraic thought
than those that attempted to solve the problem through the formulation of a system of
equations without being clear how to relate the quantities involved in the problem,
that is to say, without having recognized the structure of the problem.
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