reading levels of statistical graphs

READING LEVELS OF STATISTICAL GRAPHS Pedro Arteaga1, José Manuel Vigo2 y Carmen Batanero1, 1

Universidad de Granada (SPAIN) IES Puertas del Campo, Ceuta (SPAIN)

2

Abstract st

In this paper we present an exploratory study aimed to evaluate the reading level reached by 1 and 2nd year Vocational Training students of hairdressing and aesthetics specialty before the formal teaching of the subject. We propose a classification with five reading levels that combines those by Bertin and Curcio, and we evaluate the reading level reached by the students using this new classification. The students were given five tasks concerning different graphs (bar graphs, population pyramid, line graph, pie graph and cartogram) and were asked to reply several questions on the same. In this paper we report the results for one of these tasks. These results suggest that few students reach the upper level of critically reading the graph and the existence of a considerable proportion of students that did not reach the lower levels. We observed a better performance in the 2nd year students, who had studied some statistics the previous year. A conclusion is the need to reinforce the students’ graphical competence in order to make them able to manage in the information society. Keywords: Statistical graphs, critical reading, reading levels, vocational training students

1

INTRODUCTION

Statistical graphs are pervasive in our society and can be used to communicate information efficiently, as a tool for data analysis; therefore, the construction and interpretation of statistical graphs is also an important part of statistical literacy (Gal y Murray, 2011; Kemp y Kissane, 2010). Moreover, the study of statistical graphs is included in compulsory education at the primary and secondary school levels. Finally, in their future professional work, the students can find statistical graphs similar to those included in our questionnaire, which has been taken from statistical studies carried out in the specialty of the students of our sample. In this paper we present an exploratory study aimed to evaluate the reading level reached by Vocational Training students of the specialty of hairdressing and aesthetics before the formal teaching of the subject. Within Basic Vocational Training (MECD, 2014), mathematics does not appear as an independent subject, but is part of the Applied Sciences module. Statistical graphs appear in the section called "Graph Interpretation", where deterministic graphs are combined with statistical graphs. In particular, the textbook used by the students in the sample (Brandi, 2015) includes histograms, bar graphs, frequency polygon, pie chart and pictograms. Students are asked to represent them, to translate data into tables or graphs, to identify the statistical graph that best fits the data and activities of reading graphs. The aim of this research is performing an exploratory evaluation study to determine the reading levels of statistical graphs that reach the students of the 1st and 2nd grade in Basic Vocational Training (FPB) of the specialty of Hairdressing and Aesthetics, before formal education about the topic. Since the sample is small and the student belongs to the same educational centre, we do not try to generalize the results to another students or context.

2

THEORETICAL FRAMEWORK

Many authors have described different levels of graphical understanding; Bertin (1967) proposed the following levels: B1) Extracting data or direct reading of the data represented on the graph. For example, in a bar graph, reading the frequency associated with a value of the variable; B2) Extracting trends; being able to perceive a relationship between two subsets of data that can be defined a priori or visually in the graph. For example, visually determining the mode of a distribution in a bar graph;

B3) Analysing the data structure; comparing trends or clusters and making predictions. For example, analysing the differences in mean and range of two distributions in an attached bar graph. A related classification is due to Curcio (1989) who defined the following levels: C1) reading the data (literal reading of the graph without comparing the information contained in it), C2) reading between the data (interpreting and integrating the data in the graph), C3) reading beyond the data (making predictions and inferences from the data to information that is not directly reflected in the graph), and C4) reading behind the data, which consists on judging the method of data collection, and assessing the data validity and reliability, as well as the possible generalization of findings. In our work, to facilitate the codifying of student’s' answers we condense the two previous hierarchies into one and propose the following reading levels: •

N1: Reading the data (levels B1 or C1). When the student makes a simple reading of a data of the graph (either directly or inversely).

•

N2: Extraction of trends in a single distribution (levels B2 and C2): When it is required to compare data within a single data distribution or perform calculations with them.

•

N3: Extraction of structure in a multiple data representation (levels B3 and C2). It consists of comparing the trends of two sets of data (we can only evaluate them in graphs representing two or more distributions)

•

N4: Reading beyond the data (level C3), predicting a value that is not in the graph, i.e., interpolating or extrapolating the data.

•

N5: Reading behind the data (level C4): Giving a critical interpretation of the contents of a chart.

Research on reading levels of high school students is scarce. One of the works focussed on this subject is that of Fernandes and Morais (2011) with 108 students of 9th grade, using bar graphs, pie charts and line graphs; this was the most difficult (25.3% correct answers while in the other graphs correct responses reached 45.3%). Only 24% of students answered Level 2 questions, and 33% answered Level 3 questions in the Curcio (1989) classification, while Level 1 questions were answered by 68% of students. Pagan and Magina (2011) conducted a study with 105 students of 9th grade, based on the application of a pre-test, a classroom intervention and a post test, which includes activities of reading graphs. The authors investigated the Curcio reading levels, 67% of students reached level 1, 42% level 2 and 18.7% level 3. Carvalho, Campos and Monteiro (2011) analysed the direct and inverse reading of line graphs in English students (84 students from 7 th to 9 th degrees); 74.1% successfully did the direct reading and 37.7% the reverse, improving the results with age. Our work is based on this previous research, but uses the same graph to raise different questions in which we can reach level 4 of Curcio (level 5 in our classification); this level has not been taken into account in previous works. We also propose the classification of levels that serves to combine the categories of Bertin and Curcio.

3

METHODOLOGY

This work is carried out in a sample of 47 students, of whom only one is a boy and the remaining are girls, of two different courses (two first-year groups with a total of 29 students and one second-year group with 18 students) attending Basic Vocational Training in the branch of hairdressing and aesthetics in an educational center of Ceuta. The socioeconomic level of the students is varied and highly differentiated, both in social class, religion and previous training; the approximate age is 15-16 years old. We proposed five tasks to the students where they had to interpret different statistical graphs taken from the Internet. In this work we analyze only one task presented in Figure 1, taken from a study on consumption in hairdressing and showing three series of data: monthly average expenditure on hairdressing of men, women and in the total. The expected responses to the questions showed in Figure 1 are as follows: a. The student has to observe the variation along the time of hairdressing expenditure of men and women, that tends to grow. They can also see that the hairdressing expenditure is always higher

in women. So, they have to determine the trend in the two series and compare it. It would be a question of level N3 in our classification (level B3 in Bertin and C2 in Curcio). b. The second question asks for a prediction about a data that is not in the graph. The answer is easy, as it is enough to see that in men there is a growth of 7 euros (1.5 per year) and in women about 10 (2 euros per year). It is also possible to add this amount to the last data. Another reasonable answer would be to increase this estimated amount per year and to add the result to the last data shown in the graph. The increase in average expenditure was higher than the previous years, in both men and women, so it can be assumed that the prices of services raised. If the students respond in this way reach the level N4 in our classification (Level C3 in Curcio and not considered by Bertin). c. Although the third question is similar to the previous one, in fact it is expected that some students could observe that it is difficult to give the prediction with so much time spent without current data (7 years of difference), since the price could have changed so much and even decrease (in this case by a greater offer of hairdressers or lower consumption, due to the economic crisis of the last years). If the student responds in this way reaches the level N5 in our classification (Level C4 in Curcio´s classification).

In the following graph, it is shown the evolution of the average price in euros of the spenditure in hairdressing during the period 2004-2008. a. Was there the same evolution of the monthly average expenditure on hairdressing in men and women during the period 2004-2008? b. Based on the data shown in the graph, what do you think was the monthly average expenditure on hairdressing in 2009? c. What do you think would happen during the year 2015 in relation to the monthly average expenditure on hairdressing?

http://www.bellezapura.com/2008/03/27/datos-delbarometro-cosmobelleza-2008/

Figure 1. Task 1 presented to the students

4

RESULTS

The collected questionnaires were analyzed. For each of the student´s responses a numerical value 1 to 5 was assigned, according to the reading level reached by the student in his answer and 0 if he or she did not respond the question or if did not reach level 1 by making an incorrect reading. The answers corresponding to each item are described below, together with an example, which clarifies the way in which the responses have been coded.

4.1

First task. Evolution of average expenditure in men and women

In section a) of the item analyzed in this paper, the classification criteria of the different levels were the following:

N0: If the student does not respond to the question or if the student gives an incorrect answer. In this level, the student is not able to observe the evolution of the average hairdressing expenditure and make a comparison between men and women. N1: If the student is able to read the data but is not able to answer the question posed, being able to recognize a certain feature shown in the graph, but not related to the question; the student do not compare the information in both datasets or do not extract the tendencies. N2: The student is able to recognize the trend individually with respect to only one of the two groups (men or women), but is not able to compare both tendencies As an example, we detail the following answer, in which the student is able to observe that the monthly average expenditure in women is higher than in men, but does not get to observe the trend, only suggesting that the evolution is not the same: "No, the women spend more ". N3: If the student answers the question correctly, by clearly observing the variation in both data sets along time and observing that the expenditure is higher in women than in men. As an example, we present the following answer in which it is stated that the evolution has been the same, and the expenditure has increased globally in both distributions: "Yes, both demands increase equally".

4.2

Second task. Extrapolating a close value

In section b) of the item the classification of answers was the following: N0: If the student does not answer the question or reads the data incorrectly. Here, in the following example, the student responds with a year, when the question asks for the average expenditure, so the student is not able to establish the difference between the variables represented "2005". N2: If the student correctly reads the data, but fails to identify or compare the trends of the two data sets. In giving an approximate value but without reasoning and far away from the expected value (either in excess or default). As an example, we can show the following answer, in which the student suggests that the average expenditure will locate around two values "Around 45 to 50 €". N4: If the student responds correctly to the question posed, being able to compare the trends of two data sets and provides a possible value, taking into account the graph structure in a reasoned way. In the following example, we observe that the student gives a numerical value, which although does not identify that men’s expenditure increases 7 euros (1.5 per year) and women 10 euros (2 euros per year), he gives a reasonable estimation: "40 euros". We note that in this question the levels 1 and 3 are not considered; we do not consider level 3 because we do not ask to compare two trends and do not consider level 1 because the student who literally reads a data, usually also reaches level 2 in this question.

4.3

Third task. Extrapolation to a distant value

In the section c) the classification of answers was the following: N0: If the student does not answer the question or reads the data incorrectly. In the example, the student illustrates that he has not been able to interpret the graph, or does not understand what he is asked for: "More and more people". N3: The student reads correctly the data, being able to identify or compare the trends of the two data sets. We illustrate this level with the following example in which an obvious fact is given by the student when reading the graph, but there is no additional analysis about the situation: "Women will spend more than men" N4: If the student responds to the question giving a roughly correct value, but does not reason that it is difficult to predict without more data, i.e. the student does not reach the critical level because, although it is able to determine the graph tendency and extrapolate, he or she does not reason correctly. In the following example, the student gives an acceptable numerical data, but does not provide any reasoning: "46; get on". N5: If the student responds the question posed giving a possible value that is not in the graph and reasoning that it is difficult to predict without more current data. The student is able to determine the characteristics with respect to the monotony of the graph and in a reasonable way gives a possible value, but highlighting the difficulties of making this estimate, due to the number of years in which data are not available. This level has not been reached by any students.

Table 2. Percentage of students achieving different reading levels in the three parts of the task Grade 1 (n=29)

Grade 2 (n=18)

Part

N0

N2

N3

N4

N5

N0

N2

N3

N4

N5

a

51,7

17,2

31,1

---

---

22,2

44,4

33,3

----

---

b

27,6

44,8

27,6

---

44,4

22,2

33,3

---

c

44,8

6,9

48,3

22,2

77,8

A synthesis of the results is shown in Table 2. We can see that in part a) (evolution of the trend) the maximum level is N3, and only a third of the students reach this level. For the first-year group, more than 50% stay at N0, while most students in the second group reach N2, where students establish a correct reading of the data, and are able to establish comparisons, but not to identify the data structure, being unable to compare the two distributions. In part b) (short-term prediction), we observe something similar, although some students reach the N4 level, maximum in this question. In the second course, almost the third part of the group reaches the maximum level, while in the first group, the percentage is somewhat lower. In part c) (long-term prediction) we expected the students to reach the N5 level, but one reached that level. The percentage situated at level N4 was almost 50% in the first course and more than 75% in the second course. However the number of students who not respond or are unable of a simple reading of the graph is very high. Previous research only take into account Curcio level 3 (our level 4 (N4)), reaching 33% in Fernandes and Morais (2001), 18.7% in Pagan and Magina (2011). Our results are similar in the second question but better in the third part.

5

DISCUSSION AND IMPLICATION FOR TEACHING

Line graphs are recommended even in primary school and appear frequently in textbooks at this educational level, as shown by Diaz-Levicoy, Batanero and Arteaga (2015). Therefore, students should be familiar with this type of representation, at least for a literal reading. However, in our study many students did not achieve the first level of reading N1 that supposes only a literal reading of isolated elements of the graph. We observed better results in the interpretation of statistical graphs in the students of the second year of Vocational training in our sample, which is possibly due, on the one hand, to the degree of maturity of students in a more advanced course and in the other hand, to the education that these have acquired during the 1st course. However, the low percentage of students who reach the maximum level in each question is a matter of concern. In addition, the following errors have been repeated in the responses of the students: 

Confusing the question posed; showing little reading comprehension competence.



Not identifying the variable represented in the graph as a mean, interpreting it as a simple value of a data.



Not identifying that the average expenditure is referred to a whole year, and interpreting instead that it is referred to a concrete instant of time.



Not comparing the two distributions; although the student can analyze the evolution of each data set separately, they do not obtain a conclusion about the difference in the evolution of the two tendencies.



Not identifying the growing trend in one of the distributions.

The teacher should pay attention to the aforementioned errors and help students to achieve a sufficient level of statistical literacy that allows them to make a critical reading of the graphs they find in the media and in their professional life. Consequently, more time should be devoted to the study of statistical graphs and to performing interpretative activities. The reading of the graphs seems, at first sight, a simple activity and is taken for granted, and therefore little time is devote to its teaching; the

time is instead spend in other subjects, for example, measures of central position or dispersion. Our research and those cited in the background show that reading a graph is difficult and we must reinforce this ability in students to help them achieve a good level of statistical literacy.

ACKNOWLEDGEMENTS Project EDU2016-74848-P and group FQM126 (Junta de Andalucía).

REFERENCES [1]

I. Gal and S. T. Murray, “Responding to diversity in users' statistical literacy and information needs: Institutional and educational implications”. Statistical Journal of the International Association for Official Statistics, vol. 27, no 3-4, pp.185-195, 2011.

[2]

M. Kemp and B. Kissane “A five step framework for interpreting tables and graphs in their contexts”. En C. Reading (Ed.), Proceedings of the Eighth International Conference on Teaching Statistics. Ljubljana, Slovenia: International Statistical Institute. Disponible en http://researchrepository.murdoch.edu.au/6240/. 2010.

[3]

MECD. “Real Decreto 127/2014, de 28 de febrero, por el que se regulan aspectos específicos de la Formación Profesional Básica de las enseñanzas de formación profesional del sistema educativo, se aprueban catorce títulos profesionales básicos, se fijan sus currículos básicos y se modifica el Real Decreto 1850/2009, de 4 de diciembre, sobre expedición de títulos académicos y profesionales correspondientes a las enseñanzas establecidas en la Ley Orgánica 2/2006, de 3 de mayo, de Educación”. Madrid: Autor. 2014.

[4]

A. Brandi, “Módulo de ciencias aplicadas 1. Matemáticas 1”. Madrid: Santillana. 2015.

[5]

J. Bertin, “Semiologie graphique”. Paris: Gauthier-Villars. 1967.

[6]

F. R. Curcio, “Developing graph comprehension”. Reston, VA: N.C.T.M.1989.

[7]

J. A. Fernandes and P. C. Morais, “Leitura e interpretação de gráficos estatísticos por alunos do 9º ano de escolaridade”. Educação Matemática Pesquisa, vol. 13, no 1, pp.95-115. 2011.

[8]

A. Pagan and S. Magina, “O ensino de estatística na educação básica com foco na interdisciplinaridade: um estudo comparativo”. Revista Brasileira de Estudos Pedagógicos, vol. 92, pp. 232- 240. 2012.

[9]

C. Carvalho, T. M. Campos and C. Monteiro, “Aspectos visuais e conceituais nas interpretações de gráficos de linhas por estudantes”. Boletim de Educação Matemática, vol. 24, no. 40, pp. 679-700.2011.

[10]

D. Díaz-Levicoy, P. Arteaga and C. Batanero, “Gráficos estadísticos y niveles de lectura propuestos en textos chilenos de Educación Primaria” En C. Fernández, M. Molina y N. Planas (Eds.), Investigación en Educación Matemática XIX (pp. 229-238). Alicante: SEIEM. 2015.