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Páginas: 358-372 Recibido: 2022-01-16 Revisado: 2022-08-10 Aceptado: 2022-10-01 Publicación
Final: 2022-05-31 |
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Statistical
tables in Primary Education textbooks of Peru
Tablas estadísticas en libros de texto
de Educación Primaria del Perú
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Danilo Diaz-Levicoy
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Teresa
Sofía Oviedo Millones |
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Audy
Salcedo |
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Ximena Gutiérrez-Saldivia |
Abstract
In recent years, the literature shows an increase in research analyzing
statistical representation of data in textbooks, but in the Peruvian context is
still scarce. Therefore, this research aims to analyze activities on
statistical tables present in mathematics textbooks, published by the Ministry
of Education and distributed free of charge to teachers and students of Primary
Education in Peru. The methodology is qualitative and descriptive. For data
analysis, the content analysis technique was used by analyzing types of
statistical table, types of task, reading levels and semiotic complexity levels
as units of analysis in a complete series of mathematics textbooks of Primary
Education, from first to sixth grade, one per level, due to their wide national
coverage. The results allow us to observe predominance of tally charts and data
tables, completion and comparison tasks, reading level 2 (reading within data)
and semiotic complexity level 3 (representation of data distribution). It is
concluded that it is necessary to increase the number and variety of tasks
related to statistical tables proposed in mathematics textbooks, as well as to
reinforce the presence of the highest levels of reading and semiotic complexity
in the last years of Primary Education in Peru.
Resumen
En los últimos años, la literatura evidencia un
aumento de investigaciones que analizan representaciones estadísticas en libros
de texto, pero en el contexto peruano estas aún son escasas. Es por esto, que
esta investigación tiene por objetivo analizar las actividades sobre tablas
estadísticas presentes en los libros de texto de matemática, editados por el
Ministerio de Educación y distribuidos gratuitamente a docentes y estudiantes
de Educación Primaria en Perú. La metodología es cualitativa y de nivel
descriptivo. Para el análisis de los datos se utilizó la técnica de análisis de
contenido, analizándose las unidades de análisis, como tipos de tabla
estadística, tipos de tarea, niveles de lectura y niveles de complejidad
semiótica presentes en una serie completa de libros de texto de matemática de
Educación Primaria, de primero a sexto, uno por nivel, debido a su amplia
cobertura nacional. Los resultados permiten observar el predominio de las
tablas de conteo y de datos, de las tareas de completar y comparar, del nivel
de lectura 2 (leer dentro de los datos) y del nivel de complejidad semiótica 3
(representación de una distribución de datos). Se concluye que es necesario
ampliar la cantidad y variedad de tareas relacionadas con tablas estadísticas
propuestas en los libros de texto de matemática, así como fortalecer la
presencia de los últimos niveles de lectura y complejidad semiótica en los
últimos cursos de Educación Primaria en Perú.
Keywords / Palabras clave
Statistics, mathematics, statistics education, mathematics education, textbooks,
teaching materials, data visualization, primary education.
Estadística, matemáticas, enseñanza de la
estadística, Enseñanza de las matemáticas, libro de texto, material didáctico,
visualización de datos, enseñanza primaria.
1. Introduction
One essential
knowledge in the education of citizens, both for their social participation and
decision-making, is statistics, particularly, basic representations such as statistical
tables or graphs, which are frequently used in the media (Arteaga et al., 2011;
Gal, 2011; Jurečková & Csachová, 2020). This situation requires educational
centers to train citizens with an adequate statistical culture. This implies
that people develop the skills to:
(…)
read and interpret tables, graphs and summary measures that appear in the
media; interpret, critically evaluate, and communicate statistical information;
understand and use the language and basic tools of statistics; appreciate the
value of statistics in everyday life, civic life and professional life as a
consumer of data (Del Pino & Estrella, 2012).
From a curricular
point of view, and taking into account international trends, the National
Ministry of Education of Peru (MINEDU) has defined the national curriculum of
Basic Education - at the early, primary and secondary level - by
competence, understood as “the faculty that a person has to combine a set of
abilities in order to achieve a specific purpose in a given situation, acting
in a pertinent and ethical way” (MINEDU, 2016, p. 192). In particular, Primary
Education is made up of three stages: Stage III (1st and 2nd grade), Stage IV
(3rd and 4th grade) and Stage V (5th and 6th grade).
Mathematics area is
made up of four competences: 1) Solving quantity problems; 2) Solving problems
of regularity, equivalence, and changes; 3) Solving problems of form, movement
and location; 4) Solving data management problems and uncertainty. Teaching of
statistical tables is linked to the last competence, the purpose of which is:
(…) For
the students to analyze data on a topic of interest or study of random
situations, which allows them to make decisions, develop reasonable predictions
and conclusions supported by the information produced. To do this, the students
collect, organize and represent data that provide inputs for the analysis,
interpretation and inference of their deterministic or random behavior using
statistical and probabilistic measures (MINEDU, 2016, p. 263).
To address this
competence, which is worked on from the first year of Primary Education, it is
necessary for students to develop the following skills (MINEDU, 2016, p. 141):
1) Represent data with graphs and statistical or probabilistic measures; 2)
Communicate understanding of statistical and probabilistic concepts; 3) Use
strategies and procedures to collect and process data; 4) Support conclusions
or decisions based on information obtained. Although working with statistical
tables is implicit in each of these skills, this is only made explicit in the
second of them, where students must “read, describe and interpret statistical
information contained in graphs or tables from different sources” (MINEDU,
2016, p. 141).
These new curricular
guidelines emphasize that teaching mathematics in general, and statistics in
particular, must be focused on the use of significant situations and of
interest to students, which evoke previous knowledge and with activities of
gradual complexity. Cooperative learning and work should be promoted with the
use of different teaching methodologies: Problem-based learning, project-based
learning, case studies, among others.
Given this latest curriculum
modification, it is of interest to analyze its implementation in textbooks,
because this resource is fundamental in teaching and learning processes despite
technological development (Braga & Belver, 2016), influencing educational
practices of teachers (Olsher & Even, 2014). This pedagogical resource is
an example of the curriculum potentially taught in the classroom (Valverde,
Bianchi, Wolfe, Schmidt y Houang, 2002), as a result of a didactic
transposition (Chevallard, 1991), in which the teacher must be an epistemic
guard to avoid biases are transmitted to students (Ortiz, 2002). In addition,
there is a direct relationship between curricular guidelines and the textbook
(Herbel, 2007; Shield, & Dole, 2013), because the latter directly
influences the success or failure of their implementation (Cantoral et al.,
2015). Díaz-Levicoy, Giacomone et al. (2017) highlight the usefulness of the
textbook for teachers in the organization and development of teaching and
learning processes; it is also useful for the students themselves, because they
can consult it at any time to clarify doubts and it is in line - or so
it should be - with their cognitive development; and in the case
of families, it allows them to monitor the learning process of their children
and clarify doubts to support the educational process.
In accordance with
these considerations, the objective of this study is to analyze the activities
on statistical tables in mathematics textbooks for Primary Education in Peru.
1.1. Statistical tables
Statistical tables are considered an example of transnumeration (Wild
& Pfannkuch, 1999), in which new information must be obtained when changing
the representation system, for example, when moving from data (not grouped) to
a statistical table. For Gabucio et al. (2010) statistical tables are “one of
the specific ways of recording and organizing cognitively useful information
for a multiplicity of uses” (p. 184). Estrella (2014) defines it as:
(…) A
rectangular arrangement with a structure that comprises a set of rows and
columns […], allows to present the data corresponding to one or more variables
(characteristics of the phenomenon under study) in a classified and summarized
way, in order to display the behavior of the data and facilitate the
understanding of the information that can be extracted (p. 6).
Some elements that a statistical table contains are: 1) The Title
(informs about variables and contexts represented); 2) The Body (group of cells
in which information or data is recorded); 3) Lateral Heading (first column in
which categories of the variable are detailed); 4) Top Heading (specifies
column content below); 5) Totals (row or column showing the sum of values of
the respective cells (Estrella, 2014).
On the other hand, authors such as Cazorla et al. (2017),
Díaz-Levicoy, Morales et al. (2020),
and Lahanier-Reuter (2003) describe statistical tables used in Primary
Education: 1) Data table: Simple representation, only containing data, in which
the ideas of variable and value are used, but the ideas of frequency or
distribution are not used; 2) Tally table: Simplified version of a frequency
table, in which counts are made by means of marks or symbols within the same
cell; 3) Frequency table: Representation containing frequencies (obtained by
grouping or counting the same data) of values or categories of the variables;
4) Two-way table (or contingency table): Table in which two or more variables
are related at the same time.
1.2. Reading levels and semiotic
complexity of statistical tables
Curcio et al. (Curcio, 1987; Friel et al., 2001; Shaughnessy et al.,
1996) propose a taxonomy of reading levels with respect to statistical graphs.
This taxonomy has been widely used in textbook analysis (Díaz-Levicoy, Batanero
et al., 2016; Díaz-Levicoy et al., 2018; Díaz-Levicoy, Giacomone et al., 2017;
Jiménez-Castro et al., 2020) and adapted for the analysis of statistical
tables, as shown in other works on these representations in textbooks
(Díaz-Levicoy et al., 2015; García-García et al., 2019). The reading levels
are:
·
Level 1. Reading the data.
Literal reading of the information represented in the table.
·
Level 2. Reading within the data. When a value is obtained through comparisons or simple arithmetic
operations with data displayed in the statistical table.
·
Level 3. Reading beyond the data. It is possible to obtain a value that is not explicit in the statistical
table, and that is obtained through interpolation or extrapolation.
·
Level 4. Reading behind the data. The way of collecting the
data, the conclusions highlighted by others, the way of organizing the data,
among others, is critically valued.
Similarly, Arteaga et al. (Arteaga, 2011; Batanero et al., 2010) propose
four levels of semiotic complexity for the construction of statistical tables
and graphs, but they can be easily extrapolated to statistical tables:
·
Level 1. Representation of individual data. Table in which the ideas of variable or distribution are not used,
where isolated data, a piece of data or a portion of them are recorded, without
calculating the frequencies when appropriate.
·
Level 2. Representation of a data set, without summarizing its
distribution. Table in which all the
data are recorded, one by one, without calculating the frequencies when
necessary.
·
Level 3. Representation of a data distribution. Table in which the frequencies are recorded, prior to the grouping of the same data, working on the ideas of frequency and
distribution.
·
Level 4. Representation of several distributions in the same table. When the frequencies of two or more variables are recorded in the same
statistical table.
1.3. Background
Studies that analyze mathematics textbooks are extensive; however,
researches that perform statistical content analysis are considerably less (Batanero
et al., 2015). As a result of these studies, they have become lines of research
in Mathematics Education (Gómez, 2011) and Statistical Education (Díaz-Levicoy,
Giacomone, et al., 2016).
Regarding the analysis of statistical tables in textbooks, literature
shows an increase in recent years, but they are still scarce in the Peruvian
context. This is reported by Vidal-Henry et al. (2021), who reviewed literature
on statistical tables and graphs in Ibero-America, identifying only three
researches on statistical graphs in Primary Education textbooks in Peru (Díaz-Levicoy
et al., 2018; Díaz-Levicoy et al., 2019). In this sense, Díaz-Levicoy et al. (2018)
analyze the types of graphs, as well as the tasks, reading level and semiotic
level associated with them. Their results show predominance of bar graphs,
calculating and constructing tasks, basic reading levels, and the semiotic
level of representation of a data distribution.
Regarding statistical tables, Amorim and Silva (2016) analyze fourth and
fifth grade textbooks of Primary Education in Brazil, showing confusion with
the use of the word table, the majority presence of charts and data banks, with
predominance of the interpreting activity/skill. For their part, Evangelista
and Guimarães (2017) study activities on tables in textbooks from first to
third grade of Primary Education. The results show predominance of the table as
a type of representation, and the skills of completing the table and
interpreting data from the table.
In Chile, Díaz-Levicoy et al (2015) analyze activities on statistical
tables in first and second grade textbooks of Primary Education. This study
reports that tally tables; calculating, completing and translating tasks; the
level of reading within the data and the personal context are mostly used.
Then, Díaz-Levicoy, Ruz et al. (2017) carry out an analysis of activities with
statistical tables in third grade textbooks of Primary Education, showing
predominance of tally tables and tasks of translating, calculating and
explaining.
More recently, Pallauta et al. (2020) analyze primary mathematical
objects on statistical tables in textbooks for grades 5 to 8 of Primary
Education. The results show predominance of distribution tables associated with
semiotic complexity level 3 (representation of a data distribution). In the
field of problems, they observe activities such as table construction,
translating (changing from one representation to another), among others. The
verbal, symbolic, numerical and diagrammatic stand out. As arguments is the use
of examples or counterexamples, use of graphical representations, deductive
verbal and deductive algebraic arguments. Regarding procedures, it should be
noted that they are related to the reading and construction of tables.
Another study is that carried out by Bustamante-Valdés et al. (2021), who
analyze activities on statistical tables in Chilean mathematics textbooks for
rural schools. Their results show predominance of tally tables, reading level 2
(reading within the data), semiotic complexity level 3 (representation of a
data distribution), the task of calculating, and the personal context.
In Mexico, García-García et al.
(2019) analyze activities on statistical
tables in Primary Education textbooks (first to sixth grades). The results show
predominance of data tables, reading level 2, semiotic level 2, the task of
calculating, and the personal context. Finally, in Venezuela, Salcedo (2020)
analyzed table activities in primary and secondary education textbooks. He
found that semiotic level 3 predominates in primary school, while semiotic
level 4 predominates in secondary school.
2. Methodology
To fulfill our research objective, we use a qualitative methodology of
descriptive level. For data analysis, the content analysis technique was used (Bernete,
2014). The sample is of an intentional type and consisted of a complete series
of mathematics textbooks for Primary Education in Peru (first to sixth grades),
which MINEDU distributes free of charge to public educational centers in the
country, so they have great national coverage. In each textbook, sections in
which some type of statistical table is involved were identified to define the
following units of analysis:
·
Type of table.
Including those mentioned and described in previous research in Primary
Education textbooks (Cazorla, Magina, Gitirana and Guimarães, 2017;
Díaz-Levicoy, Morales et al., 2020; Lahanier-Reuter, 2003): 1) Data table; 2)
Tally table; 3) Frequency table; 4) Two-way table.
·
Type of task. Tasks
that can be observed in an activity are considered, that is, an activity can
include one or more tasks, in which case the corresponding number will be taken
into account. For this study we have considered those described in previous
studies (Díaz-Levicoy et al., 2015; Díaz-Levicoy, Ruz et al., 2017;
García-García et al., 2019): 1) Reading; 2) Comparing; 3) Calculating; 4)
Constructing; 5) Representing on a graph; 6) Explaining, among others, which
are described in the results section.
·
Reading levels.
Considering those described by Curcio et al. (Curcio, 1987; Friel et al., 2001;
Shaughnessy et al., 1996): 1) Reading the data; 2) Reading within the data; 3)
Reading beyond the data; 4) Reading behind the data.
·
Levels of semiotic complexity. Considering those described by Arteaga et al. (Arteaga, 2011; Batanero
et al., 2010): 1) Representation of individual data; 2) Representation of a
data set, without summarizing its distribution; 3) Representation of a data
distribution; 4) Representation of several distributions in the same table.
These units of analysis are illustrated with examples in the results
section detailed below. Furthermore, regarding the type of table and task, it
is possible to observe more than one category in the same activity; for
example, two or more different statistical tables can be included in the same
activity, therefore, all representations are taken into account.
For data collection, the instrument was designed, which includes the
units of analysis used in the study, which allowed to classify activities with
statistical tables identified in the textbooks. Finally, to ensure the
objectivity of the analysis, the classification made by each of the authors of
this research was compared, considering at least a 75 % correspondence.
3. Results
Table 1 summarizes the activities or sections of textbooks analyzed by
grade. The presence of this topic is observed in the six grades of Primary
Education, with a similar distribution per grade and with an average of 12.2
activities for each textbook.
Table 1
Frequency (and percentage) of activities
analyzed by grade
|
Grade |
Activities |
|
1st |
13 (17.8) |
|
2nd |
13 (17.8) |
|
3rd |
14 (19.2) |
|
4th |
11 (15.1) |
|
5th |
10 (13.7) |
|
6th |
12 (16.4) |
|
Total |
73 (100) |
The presence of these representations is in accordance with what is
mentioned in the Primary Education curriculum, where their work is implicitly
related to the skills established by MINEDU (2016).
3.1. Type of statistical tables
Regarding the type of statistical tables, as shown in Table 2, the
activities mainly involve three tables; tally tables (32.9 %), data tables
(31.5 %) and frequency tables (28.8%), with a slight predominance of the first
of these tables. Two-way or contingency tables are sporadically observed in the
last two years and in the first year of Primary Education.
In the analysis by grade it is observed that data tables and frequency
tables are present at all levels, while tally tables appear mainly in the first
years (first to fourth).
Table 2
Percentage of the type of table involved in
each activity
|
Type of Table |
1st (n=13) |
2nd (n=13) |
3rd (n=14) |
4th (n=11) |
5th (n=10) |
6th (n=12) |
Total (n=73) |
|
Data |
15.4 |
38.5 |
35.7 |
18.2 |
50 |
33.3 |
31.5 |
|
Tally |
53.8 |
38.5 |
28.6 |
63.6 |
0 |
8.3 |
32.9 |
|
Frequency |
15.4 |
23.1 |
35.7 |
18.2 |
40 |
41.7 |
28.8 |
|
Two-way |
15.4 |
0 |
0 |
0 |
10 |
16.7 |
6.8 |
3.2. Type of tasks
This section details the results on statistical tables that are involved
in the activities presented in the Primary Education textbooks in Peru.
The tasks identified in this study are described and illustrated with
examples below.
Reading. When the student is asked to perform a literal reading of the
information presented in the statistical table, that is, it is possible to
request the reading of a frequency, the title or a specific category. This task
is illustrated with an example in the activity in Figure 1, specifically in the
first question in section b., where students must answer what foods Miguel
brought and in order to answer this they must read the table and identify the
foods shown.
|
Miguel’s
Lunchbox Food – Tally -Total b. Answer
orally. • What food did Miguel bring? • What food did he take every day? c. Organize
the information in a graph. d. On the
chart, paint a box for each of the foods that Miguel brought in his lunchbox. e. Answer
orally, what does the longest bar represent? The food that
Miguel brought the most is____________________. |
Figure 1. Reading task example (T1, p. 61)
Comparing. When it is requested to establish some order between the data or the
information shown in the table, for example, identify the maximum or minimum
values, find those greater or less than, among others. In Figure 2, it is asked
to identify which objects were bought in greater and lesser quantity, in
addition, it is must be indicated which is the product that was bought the most
on Saturday. So the students have to compare between total quantities of
objects bought at the general level and on Saturday.
|
Mrs.
Paola bought several objects over the weekend. The chart shows the objects
that she bought. a.
Complete the table and answer. • What objects did she buy in
greater quantity? • What object did she buy in less
quantity? • What objects did she buy in greater
quantity on Saturday? |
Figure 2. Comparing task example (T1, p. 90)
Calculating. When the student must perform an arithmetic operation with the data or
information represented in the statistical table, for example, in Figure 3, it
is asked to calculate the average time that Paola achieved in her swimming
tests, to know if she manages to keep it in 15 seconds. So the students have to
calculate that average by adding the time of each test and dividing it by the
amount of these.
|
Paola’s
father helps her calculate her average time in her swimming tests. She wants
to keep it in 15 seconds. Will she have made it? Test 1 -
Test 2 -
Test 3 -
Test 4 -
Test 5 14 s -
16 s -
14 s -
17 s -
14 s a.
Calculate Paola’s average time. |
Figure 3. Calculating task example (T5, p. 82)
Completing. When it is requested to finish the construction of a statistical table
with a given structure and with the data provided in the statement (ungrouped
data, grouped in a statistical graph or in a table). An example of this task is
shown in Figure 4, where it is requested to complete the table according to the
distance measurement of the marbles in the hole shown. So the children have to
complete the table with the distance measurements of each marble with respect
to the hole.
|
Miguel and
his friends played throwing marbles. Whoever throws the marble closest to a
hole wins the game. Who won the game? Measure the distances. a. Measure
the distances and complete the table. Student/Measure a. The
winner was ____________. |
Figure 4. Completing task example (T3, p. 82)
Representing on a graph. The construction of a statistical graph is requested from the
information displayed in a statistical table. In Figure 5, from the data given
in the frequency table, the student has to finish a bar graph (following the
example given in the graph).
|
Miguel
conducted a survey among his classmates to find out what is the favorite
color in his classroom. What color turned out to be the favorite? a. Read the
table with the answers given to Miguel and paint the results on the graph. |
Figure 5. Representing on a graph task example (T1, p.
116)
Explaining. Task in which the student must argue, detail procedures, give points of
view or draw conclusions from the information represented in the statistical
table. In Figure 6, in question number five of item a, students have to explain
what they can do to find out which grade has more and less chairs. For this,
the student must mention the procedure or steps to follow to see the grade in
which there are more or less chairs.
|
a. Look at
the table and answer. • How many chairs are there in 1st grade? • How many chairs are there in 4th grade? • In what grade are there more chairs, in 2nd
or 5th? • In what grade are there more chairs, in 3rd
or 6th? • What can you do to find out which grade has more
chairs and which has fewer chairs? |
Figure 6. Explaining task example (T3, p.29)
Problem
posing. It involves creating a
situation in which the data presented in the statistical table keep sense and
coherence, as well as the formulation of questions that can be answered from
them. An example of this task is shown in Figure 7, where students have to pose
a problem, according to the information on types of race.
|
a. Write a
problem to be solved using the information in the table where what happens
most frequently is taken into account. b. Pose 3
questions and answer them. |
Figure 7. Problem posing task example (T4, p. 138)
Question posing. Students must identify or create the question that allows to obtain
some information that has been represented in the table, as well as possible
questions that could be answered with the information from the statistical
table. For example, Figure 7 shows this type of task. Students have to pose
questions from a table.
Constructing. A table must be developed with the information provided in the
statement and the student must define its structure. In Figure 8, students have
to create a frequency table from a bar graph with the following variables: food
and number of students.
|
Sixth grade
students were surveyed about the foods they prefer in their school lunchbox.
They presented the following graph. How do we find out what kind of food
sixth graders at this school prefer? a. Read the
problem and answer. • What is the problem about? • What data does the bar graph give us? • What was the survey done for? • How many students were surveyed? b. Create
the frequency table with the data from the graph and indicate the highest
frequency. |
Figure 8. Constructing task example (T6, p. 106)
Table 3 shows task distribution on statistical tables identified in the
textbooks of Primary Education in Peru. In it, it is observed that the most
frequent tasks are to complete (79.5 %), compare (46.7 %) and explain (34.2 %),
the first two present in the six grades of Primary Education. The tasks related
to problem posing (1.4 %) and question posing (1.4 %) are observed
sporadically.
Table 3
Percentage of the type of tasks related to
statistical tables in textbooks
|
Type of Table |
1st (n=13) |
2nd (n=13) |
3rd (n=14) |
4th (n=11) |
5th (n=10) |
6th (n=12) |
Total (n=73) |
|
Reading |
15.4 |
38.5 |
7.1 |
36.4 |
10 |
16.7 |
20.5 |
|
Comparing |
61.5 |
46.2 |
42.9 |
54.5 |
30 |
41.7 |
46.7 |
|
Calculating |
23.1 |
15.4 |
21.4 |
9.1 |
50 |
50 |
27.4 |
|
Completing |
92.3 |
100 |
71.4 |
81.8 |
70 |
58.3 |
79.5 |
|
Representing on a graph |
46.2 |
30.8 |
21.4 |
45.5 |
40 |
25 |
32.9 |
|
Explaining |
0 |
7.7 |
71.4 |
36.4 |
30 |
58.3 |
34.2 |
|
Problem posing |
0 |
0 |
0 |
9.09 |
0 |
0 |
1.4 |
|
Question posing |
0 |
0 |
0 |
9.09 |
0 |
0 |
1.4 |
|
Constructing |
0 |
0 |
0 |
0 |
10 |
25 |
5.5 |
3.3. Reading levels
This section describes the results regarding reading levels described by
Curcio et al. (Curcio, 1987; Friel et al., 2001; Shaughnessy et al., 1996)
identified in the activities on statistical tables in Primary Education
textbooks.
Figure 9 shows reading level 1 (data reading), where the students have to
perform a literal reading of the information given in the table, which consists
of identifying the total number of objects that will be put away by type.
|
Draw a / in the table for each material in the image and
complete the column with the total number of objects. Urpi will put away _____________; Nico
__________________; Paco ______________; and Susy ____________. |
Figure 9. Reading level 1 example (T1, p. 29)
Reading level 2 (reading within the data) is illustrated with an example
in Figure 3, where students have to read the values given in the data table and
perform a simple calculation to answer if Paola met the time of swimming she
wanted to achieve. For this the students have to calculate the average, mean or
arithmetic mean.
Reading level 4 (read behind the data) is illustrated with an example in
Figure 10, in which a situation of an election in a school council is detailed,
and the students have to relate the information given in the table with the
context, arguing whether or not there will be a runoff election, according to
the number of votes.
|
In the Educational Institution 1040 the elections of the School Council took place. The winning list must have a difference of at least 10 votes over the list that is in second place, because, otherwise, a runoff election will take place. a. Count the votes and complete the table. b. Answer. • Which list got the most votes? • Which list got second place? • Will there be a runoff election? Why? |
Figure 10. Reading level 4 example (T1, p. 29)
Table 4 shows the distribution of the reading level observed in each
activity involving statistical tables. It shows the predominance, at a general
level and by grade, of reading level 2 (reading within the data) related to the
application of algorithmic processes with the data or information observed in
the table. It is followed by level 1 (data reading) (13.7 %), also present in
all the grades analyzed. On the other hand, reading level 4 (reading behind the
data) is observed only in the third and sixth grade textbooks of Primary
Education.
Table 4
Percentage of reading level by grade
|
Reading Level |
1st (n=13) |
2nd (n=13) |
3rd (n=14) |
4th (n=11) |
5th (n=10) |
6th (n=12) |
Total (n=73) |
|
1 |
15.4 |
15.4 |
7.1 |
18.2 |
10 |
16,.7 |
13.7 |
|
2 |
84.6 |
84.6 |
85.7 |
81.8 |
90 |
75 |
83.6 |
|
4 |
0 |
0 |
7.1 |
0 |
0 |
8.3 |
2.7 |
3.4. Levels of semiotic complexity
Finally, this section describes the results regarding the levels of
semiotic complexity described by Arteaga et al. (Arteaga, 2011; Batanero et
al., 2010) in the activities involving statistical tables.
Figure 11 shows an activity involving a statistical table with semiotic
complexity level 2 (representation of a data set, without actually summarizing
its distribution). The activity consists of measuring the objects indicated in
the table and recording the measurements of length, width and total length of the
edge (in cm), so the idea of frequency or frequency distribution is not used.
|
Some classroom objects will be decorated by placing colored ribbons on their edges. a. Measure the objects listed in the table with the measuring tape on the next page and complete the table. b. Answer. • Which object has the longest edge? • What did you do to find out? |
Figure 11.
Level of semiotic complexity 2 (T3, p. 82)
Figure 5 shows an activity involving a statistical table where a data distribution
is represented (level of semiotic complexity 3), since the frequencies of the
colors preferred by Miguel's classmates are recorded.
Figure 2 shows an activity involving a two-way table, which is related to
a level of semiotic complexity 4 (representation of several distributions on
the same table). This table represents objects bought on a weekend, and
students must compare the number of objects bought according to the questions
asked.
Table 5 summarizes the distribution of activities according to the level
of semiotic complexity of the statistical table and grade. It shows the
predominance of level 3 (representation of a data distribution) (60.3 %), both
at a general level and in the first, second, fourth and sixth grades of Primary
Education. It is followed by semiotic level 2 (representation of a set of data,
without summarizing its distribution) (32.9 %) and semiotic level 4
(representation of several distributions in the same table). This last semiotic
level appears sporadically and not continuously, in the first and last two
years of Primary Education.
Table 5
Percentage of level of semiotic complexity by
grade
|
Reading Level |
1st (n=13) |
2nd (n=13) |
3rd (n=14) |
4th (n=11) |
5th (n=10) |
6th (n=12) |
Total (n=73) |
|
2 |
15.4 |
30.8 |
50 |
18.2 |
50 |
33.3 |
32.9 |
|
3 |
69.2 |
69.2 |
50 |
81.8 |
40 |
50 |
60.3 |
|
4 |
15.4 |
0 |
0 |
0 |
10 |
16.7 |
6.8 |
4.
Discussion and conclusion
Analyzing the presence of mathematical and statistical content in
textbooks is a way of approaching what is taught in the classroom, given the
relevance and use of these pedagogical resources by teachers, students and
families.
This research sought to analyze activities involving statistical tables
in Primary Education textbooks in Peru, which are published by the Ministry of
Education of Peru, considering four units of analysis: Types of tables, types
of tasks, reading levels and levels of semiotic complexity; used in previous
research.
Regarding the type of tables, tally tables are the most recurrent in
Primary Education textbooks in Peru (32.9%). These results are consistent with
those of Díaz-Levicoy et al. (2015), Díaz-Levicoy, Ruz et al. (2017), and
Bustamante-Valdés et al. (2021).
This type of table is proposed for teaching prior to frequency tables, so they
are recommended to be included in the first years of Primary Education.
Regarding the types of task, the completing task is the one that
predominates in the analyzed textbooks, with a presence in 79.5 % of the
activities. This result is consistent with previous research where this task is
among the three most frequent (Bustamante-Valdés et al., 2021; Díaz-Levicoy et
al., 2015; García-García et al., 2019). The less frequent tasks are those of
posing problems and posing questions, the latter is only observed sporadically
in the work of Díaz-Levicoy et al. (2015). On the other hand, the task of
posing a problem has been found in the study by Díaz Levicoy, Batanero et al.
(2016) on statistical graphs in textbooks from Chile and Spain.
Regarding reading levels, level 2 (reading within the data) is the one
that appears most frequently (83.6%), which is associated with the development
of calculations or comparisons, while reading level 4 (read behind the data) is
absent in four of the six grades. The predominance of level 2 is consistent
with the results of previous research (Díaz-Levicoy et al., 2015; Díaz-Levicoy,
Ruz et al., 2017; Bustamante-Valdés et al., 2021). The high prevalence of this
reading level motivates teachers and researchers to inquire about the purpose
of proposing these activities to students, so as not to reduce statistical work
to the calculation of values.
Regarding levels of semiotic complexity, the predominance of level 3
(representation of a data distribution) (60.3 %) is observed, which is
consistent with the results obtained by Bustamante-Valdés et al. (2021) and
Pallauta et al. (2020).
It should be noted that these results are novel in the Peruvian context,
since they provide background for reflection on the educational process, in
order to ensure that Primary Education students develop skills related to
statistical culture through work with statistical tables, and addressing
aspects of competence 4 (solving data management problems and uncertainty).
But, for this, it is necessary that a greater number of activities and a
variety of tasks (for example, constructing, problem posing or question posing)
be delivered on statistical tables, strengthening the work with contingency or
two-way tables in the last levels of Education Primary. In addition, it is
suggested to give greater presence to level 4 of reading and semiotic
complexity, which are sporadically observed in the activities on statistical
tables in the analyzed textbooks.
In addition, this research
provides information of interest for the training of Primary Education
teachers, who must have didactic-mathematical knowledge to teach statistical
tables. This, mainly, because the presence of this knowledge in the Primary
Education curriculum “does not ensure its teaching; that is, it is possible
that this only remains on paper” (Advíncula & Osorio, 2016, p. 1056). In
addition, various researches from Peru and other nations reveal the little
training of teachers on statistics (Advíncula & Osorio, 2016).
Finally, as a future projection, it is necessary to expand the sample of
textbooks, consider other educational levels, such as Secondary Education, and
rely on other theoretical elements. In addition, it is of interest to inquire
about the evaluative activities that teachers carry out on this knowledge in
Primary Education.
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