PROBLEMS CONCERNING VERBAL ENUNCIATION IN
GRAMMAR SCHOOL TEXTBOOKS IN MEXICO
Verónica Vargas-Alejo
CIDEM,
México
Veroalejo@hotmail.com |
José Guzmán-Hernández Cinvestav-IPN, México,
jguzman@mail.cinvestav.mx |
Abstract: The present article reports results of the analysis of verbal
problems in grammar school textbooks in Mexico. The study aims at gathering
relevant information leading to a better understanding of the kind of problems
proposed to basic students before they are formally introduced to Algebra. Our
results reveal that rate verbal problems in their various
modalities are found in most textbooks. In basic levels, a large amount of
problem information is derived from charts and designs. These problems are
notably found in fourth grade and decrease from there on. Finally, the
inclusion of verbal problems in grammar
school textbooks is sloppy.
Introduction
Our work is based on Bednarz
and Janvier research study (1996). We are, as they, interested in analyzing the
nature of verbal problems presented to students, and their relative difficulty.
This aims at providing new elements for a better understanding of students’ transition to Algebra. We are
particularly concerned with grammar school education as a direct background,
where Arithmetic provides the students with a variety of procedures they may
put to action when faced with “algebraic” problems in highschool.
We intend to find out
whether there is a spiral phenomenon in the design and structure of verbal
problems in grammar school Mathematics textbooks; if there are general
patterns maintained in the problems
presented in these books; we also intent to become acquainted with the elements
that are incorporated or omitted in problems which makes them ever more complex,
as well as to identify advantages and disadvantages of problems presentation
array. We may hereafter make a proposal
to facilitate the teaching of such verbal problems.
1. Theoretical framework
Among the
diverse approaches supporting meaningful learning of Algebra is Bednarz and
Janvier’s perspective on problem-solving (1994) that studies the transition
from Arithmetic to Algebra in highschool students. Bednarz and Janvier consider
that the surfacing of algebraic thinking in a problem-solving context brings
about reflection on the nature of the problems presented to the students in
both the algebraic and the arithmetical domains and on the solving procedures
that come to hand. In this respect, their research focuses on the setting up of
a theoretical instrument allowing for a classification of problems used in the
teaching of Arithmetic and Algebra. The inherent purpose of this theory is to
understand students’ difficulties when solving verbal enunciation problems and
to understand as much as possible their reasoning in solving them. The theory
makes prediction of problem complexity possible, whether arithmetical or
algebraic.
Bednarz and
Janvier (ibid) have created a symbol system to support their theory. Their
purpose is to separate the mathematical-relational structure from the context elements.
In other words, identifying the nature of data (known and unknown), the
relations among them and the structure of such relations. For example, addition problems, that have to
do with transforming. The symbolism used by Bednarz and Janvier is the following:
known quantities are symbolized by black squares; unknown quantities are
symbolized by white squares. A question mark is placed over the white squares
representing the problem’s unknowns to be unfolded. There are four types of
relationships among quantities: comparison,
operation, transformation and rate.
The symbolism representing them is the following:
|
|
operation |
transformation |
Rate |
The sign
"*" indicates the position of one of the operations: addition,
subtraction, division and multiplication. The symbol = means that there is a
comparative relation between quantities.
On the basis of
this symbolism, Bednarz and Janvier analyzed the verbal enunciation problems
-arithmetical and algebraic- in different Mathematics textbooks at various
educational levels.
The three kinds of problems they identified were: rate, transformation and
unequal distribution. The rate
problems are those involving comparison between non homogeneous magnitudes; transformation problems involve one or
several amount or magnitude transformations, and in unequal distribution they deal with the whole-and-parts problem.
2. Methodology
The theoretical
construct supporting this study was developed by Bednarz and Janvier (1996).
All of the verbal problems in six grammar school textbooks were classified;
problems were analyzed in terms of the known and unknown quantities, the
relation between them and the kind of relationship involved, and they were
classified according to the three problem categories that were identified: rate, transformation and unequal distribution. Finally, the
problems presentation array is explained in terms of their structure. We may
therefrom implement and experiment a didactical proposal focused on the
improvement of the presentation array of diverse verbal problems, in order to
give further didactical support to teachers.
3. Advances
The research process is
currently dealing with the identification and analysis of verbal enunciations
in grammar school textbooks. We have identified 234 verbal problems altogether
which are distributed the following way: 7 % in first grade (1°), 22 % in
second grade (2°), 16 % in third grade
(3°), 26 % in fourth grade (4°), 17 % in fifth grade (5°) and 12 % in sixth
grade (6°). Condensed information appears in the following graph:
Fig. 1
We have found
that a high percentage of verbal problems (97% approximately) is arithmetical
or connected. That is, the student
may “build bridges” among the problem’s data in order to unfold the unknowns
and, then, solve the problem arithmetically. There is no need to work on more
than one state simultaneously. When one is obtained, the rest can be obtained.
For example: If Ruben took 8 minutes to
get from his house to the football field on his bike and the game started right
after, how long was the game if you know that he burnt 208 calories altogether?
(The problem explains that six calories are burnt every minute when riding a
bike and 8 calories per minute are burnt playing football, (5th grade problem,
page 83, see fig. 2)
Fig.
2
This scheme
shows the kind of relations between
known and unknown quantities and the problem conditions. On the left you can
see the known quantities; Bednarz and Janvier’s symbolism (Ibid) is on the
right. “Stressed” rectangles show the problem’s data.
The other 3% are
the so called disconnected problems,
which do not allow for “bridges” among the known quantities. Various states
must be operated at the same time, that is, the utilization of an equation
becomes necessary. As a consequence, the student ends up exerting algebraic
type of thinking. (See fig. 5.)
As to the
general structure of the identified problems, we have classified them as transformation rate problems and unequal distribution problems. 95% of
all of the problems are the rate kind.
For example: How many times can a 10 m
long rope fit in a 100 m2 football field? (5th grade problem, page 45).
Fig.
3
Transformation problems appear in a low
percentage, 4% approximately. They are found in the last three grades of
grammar school. For example:
The mountain gorilla walks with flexed legs
measuring 1.75m. It is 0.25m taller when stretching its legs. The giraffe is
8.5m tall. What is the difference between the giraffe’s size and the gorilla’s
with stretched legs? (4th grade problem, page 160)
Fig.
4
And, finally,
there is only one problem involving unequal distribution:
“I’m glad you mention it because the bus has
18 double seats only and Mrs Dominguez and Mrs. Rocha and myself are going too
to see to the group. We are 48 people altogether...” How many people are there
in Chela’s group? If there are 5 more
girls than boys, how many girls and how many boys are there? (5th grade
problem, page 61)
Fig.5
Given that rate problems appear the most in grammar school textbooks, they were analyzed
in terms of the unknown’s placement. The rate is known in 197 out of 223 rate
problems, whereas in the 26 remaining problems (12% approximately), the unknown
is the rate. 43% of the 197 rate problems have the following scheme:
Fig.
6
Problem example: How many legs must Carlos cut out to make 5
bears out of them? (2nd grade problem, page 56.)
81% of the 26 rate problems
having the rate as the unknown have the following scheme:
Fig.
7
Example of this
kind of problems: “I live in a 30 house little village. After the class, I walk
to the parcel which is 2 km away from the village to take food to my father.
Once I am back, I play with my friends.” If an average of 180 people live in
the village, how many people average live in a house? (6th grade problem, page
81.)
Scheme distribution and
their differences are the following: Most of the first grade problems are of
the “adding rates” type. That is, 71%
of the schemes are of the fig. 2 type. In second grade, the scheme type varies
very little. 32% are of the fig. 2 type, but here the operation is not only
adding but subtracting; and 32% are of the fig. 6 type. In third grade,
problems are again focused on the two previous schemes. 41% are of the fig. 6
type, 15% are of the fig.2 type and a 10% are represented in fig. 7. In fourth
grade, 41% of the problems focused again on fig. 6, 11% on fig. 7 and 13% on
fig. 2. In 5th grade, 33% of the schemes are of the fig.6 type. Finally, in 6th
grade 55% of the problems follow the fig. 6 scheme. The schemes of the
remaining rate problems are different
from one another in general and do not necessarily reappear by grade.
4. Conclusions
As a
result of this advance we may conclude that, from a theoretical perspective,
the presentation of verbal enunciation problems in textbooks has not been taken
care of, and this happens in the first place because their occurrence is not
frequent; most problems are found in fourth grade and the number decreases from
there on. On the other hand, problems are concentrated mainly in three types of
schemes: those represented by figure 2, figure 6 and figure 7. What we have
observed in the verbal enunciation problems we analyzed is the growing
complexity of the quantities dealt with and the contexts in which they appear.
In the first grades, students may resort to illustrations for support and in
the last grades they must derive most information from the texts. On the other
hand, the fact that we are in an arithmetical context does not necessarily
imply that the problems presented to students must be reduced only to the so called “connected” problems.
Introducing oneself into the solution of “disconnected” problems is also
important.
References
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4th grade. Mexican Ministry of Education (SEP).
Ávila, Alicia et al (1994). Mathematics.
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Bednarz, N. y Janvier, B. (1994). The emergency and development of
algebra in a problem solving context: a problem analysis. PME. Lisbon.
Portugal.
Bednarz, N. y Janvier,B. (1996). Emergence and development of algebra as a problem solving
tool: continuities with arithmetic.
CIRADE. Université du Québec a Montréal.
Block, David F. Et al (1994).
Mathematics. 1st grade. Mexican Ministry of Education (SEP).
Fuenlabrada, Irma R. Et al (1997).
Mathematics. 2nd grade. Mexican Ministry of Education (SEP).
Guzmán, J. Bednarz, N. y Janvier, B. (No publicado). Solving verbal enunciation algebraic problems: In search of teaching alternatives.
Internal discussion document.
López, Gonzalo et al (1994). Mathematics.
6h grade. Mexican Ministry of Education (SEP).
Pérez, Esnel et al (1994). Mathematics.
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