SASHI VARTA SHARMA
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND
GRAPHS: IMPLICATIONS FOR RESEARCH
ABSTRACT. Concerns about students’ difficulties in statistical reasoning led to a study
which explored form five (14- to 16-year-olds) students’ ideas in this area. The study
focussed on descriptive statistics, graphical representations, and probability. This paper
presents and discusses the ways in which students made sense of information in graphical
representations (tables and bar graph) obtained from the individual interviews. The
findings revealed that many of the students used strategies based on prior experiences
(everyday and school) and intuitive strategies. From the analysis, I identified a fourcategory rubric for classifying students’ responses. While the results of the study confirm
a number of findings of other researchers, the findings go beyond those discussed in the
literature. While students could read and compare data presented in a bar graph, they
were less competent at reading tables. This could be due to instructional neglect of these
concepts or linguistic and contextual problems. The paper concludes by suggesting some
implications for researchers.
KEY WORDS: beliefs, experiences, high school students, interpretation of tables and
graphs, interviews, statistical thinking
In recent years statistics has gained increased attention in our society.
Decisions concerning business, industry, employment, sports, health, law
and opinion polling are made using an understanding of statistical information (Scheaffer, 2000). Paralleling these trends, there has been a
movement in many countries to include statistics at every level in the
mathematics curricula. In western countries such as Australia (Australian
Education Council, 1991), New Zealand (Ministry of Education, 1992)
and the United Kingdom (Holmes, 1994) these developments are
reflected in official documents and in materials produced for teachers.
In line with these moves, Fiji has also produced a new mathematics prescription at the primary level that gives more emphasis to statistics at this
level (Fijian Ministry of Education, 1994). The use of relevant contexts
and drawing on students’ experiences and understandings is recommended for enhancing the students’ understanding of statistics (Gal, 1998;
Watson, 2000; Watson & Callingham, 2003). For instance, Watson and
Callingham write that Bstatistical literacy is incomplete without the
opportunity to engage with genuine social contexts, particularly such as
International Journal of Science and Mathematics Education (2006) 4: 241Y268
# National Science Council, Taiwan 2005
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SASHI VARTA SHARMA
those found in media items^ (p. 21). Clearly the emphasis in these
documents is on producing intelligent citizens who can reason with
statistical ideas and make sense of statistical information.
Lajioe & Romberg (1998) write that in spite of its decade long
presence in curricular reform in mathematics education, statistics is an
area still in its infancy. Research shows that many students find statistics
difficult to learn and understand in both formal and everyday contexts
and that we need to better understand how learning and understanding
may be influenced by ideas and intuitions developed in early years
(Biehler, 1997; Friel, Curcio & Bright, 2001; Gal & Garfield, 1997;
Mevarech & Kramarsky, 1997; Shaughnessy & Zawojewski, 1999).
Most of the research in statistics has been done at the primary levels or
with tertiary level students. Although there is a growing body of research
at the lower secondary level there still appears a gap in our knowledge
about students’ conceptions of statistics at the upper secondary level.
Additionally, for many teachers statistics continues to be a content area
in which they have little experience since it has only recently become a
core area on some curricula. And perhaps more traditional mathematics
teaching skills do not transfer into this new domain. This may be due in
part to the fact that mathematics is so often taught as a subject focused
on procedures. In statistics surely it is even more helpful than in other
areas of mathematics to place the emphasis on helping students learn to
formulate questions, gather data, and use data wisely in solving real
problems. In order to help inform teachers and curriculum designers, it
appears to be crucial to carry out investigations at the secondary level.
Concerns about the importance of statistics in everyday life and in
schools, the lack of research in this area and students’ difficulties in
statistical reasoning, determined the focus of my study. Overall, the
study was designed to investigate the ideas Form Five students have
about statistics (descriptive statistics, graphical representations and
probability), and how do they construct these. This paper presents and
discusses data obtained from the graphical representation tasks (tables
and bar graph). I shall briefly mention the theoretical framework and
some examples of the work focusing on the graphical representation data
by students before turning to the results of my own study.
THEORETICAL FRAMEWORK
Much recent research suggests that socio-cultural theories combined with
elements of constructivist theory provide a useful model of how students
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
243
learn mathematics. Constructivist theory in its various forms, is based on
a generally agreed principle that learners actively construct ways of knowing as they strive to reconcile present experiences with already existing
knowledge (von Glasersfeld, 1993). Students are no longer viewed as
passive absorbers of mathematical knowledge conveyed by adults; rather
they are considered to construct their own meanings actively by reformulating the new information or restructuring their prior knowledge (Cobb,
1994). However, this active construction process may result in alternative views as well as the student learning the concepts intended by the
teacher. The research on students’ interpretation of tables and graphs
(Curcio, 1987; Estepa, Bataneo & Sanchez, 1999; Gal, 1998; Shaughnessy
& Zawojewski, 1999) provide evidence of the construction process. The
researchers report that students are likely to have beliefs about the features
of tables and graphs that are different from what is expected.
Another notion of constructivism derives its origins from the work of
socioYcultural theorists such as Vygotsky (1978) and Lave (1991) who
suggest that learning should be thought of more as the product of a social
process and less as an individual activity. There is strong emphasis on
social interactions, language, experience, collaborative learning environments, catering for cultural diversity and contexts for learning in the
learning process rather than cognitive ability only. Research on graphing
(Cobb, 2002; Roth, 2004) indicate that a socioYcultural perspective
towards graphing may avoid the deficiencies of a cognitive model.
Mevarech & Kramarsky (1997) claim that the extensive exposure of our
students to statistical graphs outside schools may create a unique situation
where students enter the mathematics class with considerable amount of
graphing knowledge. This means that during the teaching and learning
process, students draw inferences about the new information presented to
them by relating to some aspect of this prior knowledge to develop a deeper
meaning for statistical concepts. This research was therefore designed to
identify students’ ideas, and to examine how they construct them.
RESEARCH
ON
ANALYSIS
AND INTERPRETATION OF
TABLES
AND
GRAPHS
Curcio & Artzt (1997) state that the ability to interpret and predict from
data presented in graphical form is a higher-order skill that is important
in our technological society. Curcio and Artzt’s focus on graphical representations is consistent with the views of Gal (2002), Gal & Garfield
(1997) and the Ministry of Education (1992). A stated aim of the Mathematics in New Zealand Curriculum (Ministry of Education, 1992), for
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SASHI VARTA SHARMA
example, is to provide opportunities for students to interpret data in
charts, tables, and graphs of various kinds. Gal & Garfield (1997) suggest
that since most students are more likely to be consumers of data rather
than researchers, they need to learn to interpret results from statistical
investigations and pose critical and reflective questions about reported
data. Clearly reading, interpreting, and using data from tables and graphs
are important cognitive skills for all. Pfannkuch & Rubick (2002) write
that little research appears to have been conducted on students’
construction and interpretation of statistical data tables. This claim is
consistent with the views expressed by Friel, Curcio & Bright (2001). All
these authors add that this lack of research raises questions about how
students perceive data presented in data graphs and tables.
Biehler (1997) analysed senior secondary students’ efforts to compare
data sets. The students in this study were presented with raw data sets
and the software Datascope and asked to produce any appropriate
summaries or graphs. Biehler found that students frequently dealt with
messy tabular data than thinking that they would see more structure in
the data through graphical representation. In addition, he was concerned
about students’ abilities to interpret information, express it verbally and
consider alternatives. Estepa, Bataneo & Sanchez (1999) studied how
senior secondary students compared two small data sets with numeric
values presented in tables. One of the questions was based on ten beforeYafter measurements of blood pressure involving a medical treatment,
and the other was based on blood sugar levels of 10 boys and 10 girls.
These authors classified the students’ strategies into three categories;
correct, partially correct and incorrect strategies. Some of the correct
strategies were: comparing means, comparing totals, and comparing the
two distributions. The partially correct strategies included finding out the
difference for all pair wise cases and taking into account exceptional cases.
Three incorrect strategies included comparing highest and lowest values,
comparing ranges and basing conclusions on previously held theories or
beliefs about the context of the data set.
A number of research studies from different theoretical perspectives
seem to show that students are particularly weak in drawing inferences
and predicting from graphs (Asp, Dowsey & Hollingsworth, 1994; Bright
& Friel, 1998; Curcio, 1987; Pereira-Mendoza & Mellor, 1991). Curcio
(1987) studied graph comprehension strategies of fourth and seventh
grade students and identified three levels of graph comprehension which
relate to the kinds of tasks graphs can be used to address. These levels
are: reading the data involving simple extraction of values, reading
between the data involving comparing values, and reading beyond the
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
245
data. Curcio suggested that children should be actively involved in
collecting the data and then describe the relationships and patterns that
they observe in the sets of data that they have collected. These extended
experiences will help students to read data competently.
Meanwhile, other writers have recognised that for students to effectively utilise graphs it is not sufficient for them to just be able to directly
read information from a graph; they should be able to analyse and interpret
from data and graphs in social settings. For instance, Pereira-Mendoza
(1995, p. 2) notes:
While drawing a graph and answering factual or low level interpretative questions are
components of developing graphical skills, these are not the only components. In fact, in
terms of the ability to use graphs in problem-solving situations or to analyse critically
data in newspapers, on television or in other documents, these are the least important
components.
The above issues were addressed in some detail in the PereiraMendoza & Mellor study (1991). Here 121 grade 4 students and 127
grade 6 students were questioned on 12 different graphs, covering real/
familiar topics such as height of children. Each graph consisted of three
questions, a literal question, an interpretation question, and a prediction
question. Although there were very few problems with the literal reading
of graphs, there were major problems with the interpretation questions
and the prediction questions. The analysis indicated two main sources of
errors: data arrangement and the fact that the information was not shown
on the graph. Another common difficulty students experienced was going
beyond the graph; they could not give an answer because the information
was not on the graph. Similarly, Asp et al. (1994) described a preliminary
study into primary and post-primary students’ understanding of pictographs and bar graphs. They reported that students had fairly welldeveloped skills in reading, interpreting and predicting from graphs, and
that these increased with ability level and peer level. Nevertheless, the
students still experienced difficulty related to prior knowledge, missing
data, scale and pattern.
Bright & Friel (1998) conducted a study of the ways that students in
grades 6, 7, and 8 make sense of information in bar graphs. They explored ideas of reading the data, combining and comparing graphs and
predicting from data on the following task.
A class of students has been collecting information about themselves. One question that
they wanted to find out was how many children each person in class had in his/her
family.
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First, the subjects were shown a graph of unordered data and asked to
find out how many children were in all the families in the class altogether. Next, the researchers gave them a new graph (Figure 1) and
asked their subjects to study this new graph for a few minutes and
describe how they can find out how many students are in the class. The
researchers report that although these students had been exposed to many
different bar graphs during both in-school and out-of-school experiences,
they were not highly successful at answering questions that required
higher order thinking skills. The students tended to want to move quickly
to manipulation of information or seemed to be interpreting graphs in
ways that were inconsistent with clear understanding of the underlying
principles. The researchers claim that the process of data reduction and
the structure of graphs are factors that influence graph knowledge. Bright
& Friel (1998) argue that tables may play an important role as an
intervening representation that can smooth the transition between representing raw and reduced data.
Summary of Findings
Mathematics educators should realise that the students they teach are not
new slates waiting to have the formal theories of statistical reasoning
written upon them. The students already have their own conceptions and
beliefs about statistics and these cannot, as it were, be simply wiped
away. If student conceptions are to be addressed in the process of
instruction, then it is important for teachers to become familiar with the
alternative conceptions that students bring to classes. On the other hand,
given the subtleties of interpretations that have been reported, it is
unlikely that the research items used in the research described in this
section would have discriminated finely enough. Some of the nonstatistical responses addressed in the research literature may actually be
Figure 1. Interview question to prove understanding of bar graph.
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
247
due to misinterpretation of the questions and students’ lack of experience
at explaining answers. Additionally, students’ difficulty at explaining
may have resulted from the researchers’ own lack of clarity about how
they expected students to be able to talk about the graphical representation and their corresponding lack of clear criteria for what they
expected students to say (Friel Bright, Frieerson & Kader, 1997).
Moritz (2003) writes that many tasks to interpret graphs used by
researchers involve numerical answers, such as reading values, comparing values, and predicting values. Watson (2001) suggests a research
design that involves both visual and numerical presentation of data to
explore if the original presentation of data influence the techniques that
students use for their analysis. Interpretation from tables and graphs
include not only numerical interpretations, but also opinion statements,
such as those suggested by Gal (1998; 2002) and Pereira-Mendoza
(1995). In order to help inform teachers and curriculum designers, it
appears to be crucial to carry out investigations at the secondary level.
Such information may help teachers plan learning activities and students
overcome their difficulties. In the current interview-based study, both
literal reading and opinion questions were used to determine specific
student ideas and the factors that contribute to these constructs. The
focus on data sense is due to the fact that this is a vital part of being able
to use statistics in real world. An overview of the research design
follows, after which I will discuss the results of my study.
OVERVIEW
OF THE
STUDY
The secondary school selected for the research was a typical Fijian high
school. The sample consisted of a class of 29 students aged 14 to 16 years
of which 19 were girls and 10 were boys. According to the teacher, none of
the students in the sample had received any in-depth instruction on
statistics prior to the first interviews. The whole class participated in the
first phase of interviews, and due to time constraints 14 students participated in the second phase during which the class teacher taught a unit
on statistics and probability (see Form V of Appendix 1). This group of
14 was representative of the larger group in terms of abilities and gender.
The data reported in this paper comes from the second phase interviews.
Tasks
To explore the full range of students’ thinking about graphical representations, open-ended questions to do with tables and bar graph were
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SASHI VARTA SHARMA
selected and adapted from those used by other researchers. The task
comparing temperatures of Ba and Sigatoka (Item 1) was used to elicit
the students’ ideas about reading and interpreting tables. It involves a
context which was familiar to students. The particular numbers chosen
for the table were hypothetical data that I happened to come across and
hence each day, the temperature for Ba is higher than for Sigatoka. The
students were not told whether the given temperatures were taken at the
same time of day, or were maxima or minima. This was done deliberately to see whether students would consider the validity of the data.
The first question about whether Ba is warmer than Sigatoka was used to
explore if students could read tables. The second question, What else do
these figures reveal about the temperatures in Sigatoka and Ba?
attempted to find if students could interpret tables. Students could draw
conclusions about the temperatures of the two towns by comparing appropriate measures or finding patterns. For instance, it got colder over
the first three days in both towns and then warmed up on the next two
days. In contrast to the first question, this question requires engagement
with the overall pattern of data and consolidation of mathematical and
statistical understanding.
Item 1: Task Comparing Temperatures of Ba and Sigatoka. Temperatures (in degrees C) were taken from Sigatoka and Ba on six consecutive
days.
(1) Look at the temperatures from both the towns and decide if Ba is
warmer than Sigatoka. How do you know? Can you explain your
answer?
(2) What else do these figures reveal about the temperatures in Sigatoka
and Ba? Can you summarise the information in another way? Can
you explain your answer?
Day
Sigatoka
Ba
1
2
3
4
5
6
25
28
24
27
21
26
20
25
23
29
24
30
The graph relating to the height of several Sharma children (Item 2)
was used to examine students’ understandings of bar graphs. This context again was familiar to the students, in that it concerns the heights of
four children in a family. The first question was designed to explore
student ability in literal reading of bar graphs. The question required
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
249
students to lift numbers from specific locations in the graph or compare
two such numbers. In explaining their answers to this question, students
had to simply point to a data point in the graph. In short, answers to
literal reading questions are simple and can be unambiguously classified
as either right or wrong. The second question explored student ability in
answering questions requiring higher order statistical skills. Responses
demand qualitative descriptions rather than numerical answers.
Item 2: Height of Sharma Children. The following graph shows the
height of four of the Sharma children, ages 4, 8, 13, and 19.
(1) How tall is the 4-year-old? How do you know? How much shorter
is the 4-year-old than the 19-year-old? How did you work that out?
(2) A fifth child in the family is 10 years old. Can you tell how tall the
10-year-old is? Explain your answer.
The appropriateness of the above interview tasks for the Fijian students was established by checking the tasks with the class teacher and the
HOD mathematics at the school.
Interviews
Each student was interviewed individually by myself in a room away
from the rest of the class. During the interview, care was taken to avoid
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leading the students towards any particular viewpoint, so responses to
questions were accepted as they were given and probing questions were
asked simply to clarify the reasons for what the student thought. I
believed that this approach would allow students to demonstrate
statistical understanding and questioning which would not have been
possible in a multiple-choice format. The interviews were tape recorded
TABLE I
Characteristics of the four categories of responses
Response type
Comparing temperatures (Item 1)
Comparing heights (Item 2)
Non-response
Complete silence, I don’t know,
I have forgotten the rule
(Both questions)
Complete silence, I don’t know,
I have forgotten the rule
(Both questions)
Non-statistical
responses
Refer to everyday and school
experiences or make
inappropriate connections with
other learning areas
Linguistic and reading problems
Could not explain response
(Both questions)
Repeat previous response (Q2)
Vague or incomplete responses
Refer to everyday and school
experiences or make
inappropriate connections
with other learning areas
Linguistic and reading problems
(Both questions)
Partial-statistical
responses
Refers to one or two data points
Compare columns
(Both questions)
Inconsistent reasoning (Q2)
Adapted the rules or applied
them inappropriately (Q2)
Read heights accurately but
could not compare values (Q1)
Adapted or applied rules
inappropriately (Both questions)
Inappropriately forcing a
pattern on the graph or
alternatively not identifying a
pattern when it existed (Q2)
Statistical
responses
Pointing to both rows (Q1)
Able to generalise
all information (Q1)
Comparing mean or range (Q2)
Comparing highest and lowest
temperatures (Q2)
Finding patterns-temperatures
steadily increase (Q2)
Comparing differences (Q2)
Read and compare values with
accuracy and point to the
appropriate data point (Q1)
Refer to the height of the
8-year-old and the 13-year-old
and express the answer in a
provisional way using words
like about, between or
probably (Q2)
Recognise that information is
not given in the graph (Q2)
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
251
for analysis. Each interview lasted about 40 to 50 min. Paper, pencil and
a calculator were provided for the student if he or she needed it.
ANALYSIS
OF
DATA
The data analysis was conducted using the transcripts which were read
and re-read by myself. Analysis of the transcripts indicated that the students used a variety of intuitive strategies to explain their thinking. The
data also revealed that many of the students held beliefs and used strategies based on prior knowledge. I created a simple four category rubric
that could be helpful for describing research results relating to students’
statistical conceptions, planning instruction and dissemination of findings
to mathematics educators. The four categories in the rubric are: non
response, non-statistical, partial-statistical and statistical. These are described in Table I. The non-statistical responses were based on beliefs
and experiences while the students using the partial-statistical responses
applied rules and procedures inappropriately or referred to intuitive
strategies. The term statistical is used in this paper for the appropriate
responses. However, I am aware that such a term is not an absolute one.
Students possess interpretations and representations which may be
situation specific and hence these ideas have to be considered in their
own right. Statistical simply means what is usually accepted in standard
mathematics text-books in Fiji. It would be reasonable to assume another
category (advanced-statistical), equivalent to Shaughnessy’s (1992) pragmatical statistical level, where students appear to have a very complete
view that incorporates questioning of data but the need for such a category did not arise in my research and any responses that could have
been categorised as advanced-statistical were simply grouped with the
statistical responses.
RESULTS
AND
DISCUSSIONS
This section reports data on students’ understanding of tables and bar
graphs. The main focus is on the non-statistical responses (in which
students made inappropriate connections with learning in other areas)
and the partial-statistical responses (in which students applied rules
and procedures inappropriately, referred to some data points without
generalising to all information or forced patterns on data). In each of
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these sections the types of responses are summarised and the ways in
which the students explained their thinking is described. Extracts from
typical individual interviews are used for illustrative purposes.
Throughout the discussion, I is used for the interviewer and Sn for
the nth student.
Interview Responses About Interpreting Tables
Response types for the comparing temperatures task (Item 1) are summarised in Table II.
Table II reveals that while there were two non-responses for the
interpretation question, there were no non-responses for the reading task.
At the statistical end, while nine students could read tables, only four
could both read and interpret tables. One possible explanation for these
differences in reading and interpreting tables could be a lack of emphasis
in classrooms on interpreting tables (see Appendix 1). Since the students
lacked experiences in interpreting tables, they were more likely to use
the non-response and non-statistical categories. The students who were
classified statistical on the interpretation task considered all data points
and were also able to draw valid conclusions from the data using appropriate measures. For instance, one student stated that the highest
temperature in Sigatoka was the same as the lowest temperature in Ba.
Non-Statistical Responses. Five students used non-statistical responses
for both the questions. The non-statistical category consisted of students
who mostly related the data to their everyday experiences in nonstatistical ways. Of the five students who gave non-statistical replies on
TABLE II
Response types for the task comparing temperatures (n = 14)
Number of students using it
Response type
Task (1)
Reading tables
Task (2)
Interpreting tables
[Both tasks]
Non-response
Y
2
Y
Non-statistical
Partial-statistical
5
Y
5
3
[5]
Y
Statistical
9
4
[4]
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
253
the first task, four based their reasoning on their everyday experiences.
The students said Ba was warmer than Sigatoka and when asked to explain their answer talked about the everyday weather conditions of Ba
and Sigatoka. Student 21 appeared to be inconsistent in his explanations.
Each day the temperature for Ba is greater than the temperature for Sigatoka. And
normally Sigatoka is called a valley. They are producing fruits; it rains there. My one
uncle is there. He mainly plants Chinese cabbages; because of the rain it grows so well
there.
The initial answer (in the first sentence) is a reasonable response.
However, in the next sentence the student goes off tangent. The student
ignores the thrust of the question, which is about comparing temperatures
of two towns and instead talks about weather conditions.
Student 9 said Sigatoka was warmer than Ba because the temperatures for Sigatoka were lower than the temperatures for Ba. Even when
questioned, the student did not change her explanation. It is possible that
the student had language difficulties. She may have confused warm with
cold. However, I did make an attempt to explain the terms in vernacular
as well as in English by using an example of cold and warm water. When
asked to explain what else the figures revealed about the temperatures in
the two towns, student 9 continued to talk about Sigatoka being warmer
than Ba because every day the Sigatoka temperature was lower. The
other four students continued to base their reasoning on everyday experiences. For example, student 3 explained,
Yes, because as I told before Sigatoka is a rainy place. It hardly rains in Ba.
Although this study provides evidence that reliance upon previous
experience can result in biased, non-statistical responses, in some cases
this strategy may provide useful information for other purposes. For example, student 21’s knowledge of geography may have been reasonable.
The student has drawn on relevant common sense information. Gal
(1998) suggests that such opinions constitute what students know about
the world, they cannot be judged as inappropriate until a students’ assumptions about the context of the data are fully explored. The responses
raise further questions. Is there a weakness in the wording of this task in
that it is completely open-ended and does not focus the students to draw
on other relevant knowledge? For instance, the item involves a context
which would allow students to reach the same conclusion using the data
about temperatures in the table or using their everyday knowledge about
the temperatures in two familiar towns. The wording of part (1) of the
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SASHI VARTA SHARMA
task is not very explicit either. For example, the students were not told
whether the given temperatures were taken at the same time of day, or
were maxima or minima. This lack of information could have prevented
students drawing on tabular data because the data set did not make sense
to them.
Partial-Statistical Responses. Students’ responses that were classified as
partial-statistical on the task involving interpretation of tables (What else
do these figures reveal about the temperatures of Sigatoka and Ba?) simply repeated the responses they had given for the first item. Some looked
at the data and made some type of visual comparison. For example, student 17 said,
Ba is warmer . . . Just by looking at the numbers, just to say that this is more than
Sigatoka.
The student chose one of the temperatures in Ba and compared it with
one of the days in Sigatoka. It seems the student noticed some features of
the table yet had trouble integrating them into a coherent picture, and so
chose to report about only a few of the details examined. A possible way
of using the table would be to take account of all the temperatures it
contains. For example, the student could have compared the differences
in temperatures in Ba and Sigatoka over the period and noted a steady
increase in the differences. It must be noted that the wording of this
question may have confused some students. The students were not sure
of what to do, whether to apply statistical techniques or draw some
conclusions and hence they did not respond statistically.
Interview Responses About Reading and Predicting from Bar Graphs
Responses types for the bar graph (Item 2) are summarised in Table III.
Unlike Item 1, none of the students’ responses were classified in the
non-statistical category for the bar graph. In part (1) students were asked
to read off heights from the graph. This is a fairly standard mathematical
task and does not require students to draw on other knowledge. Moreover,
the context for Item 2 concerns a fictional family so students cannot make
use of prior knowledge about specific family members. Although the
responses of all 14 students were classified as statistical on the first
question, only five students did not attempt to impose a pattern or give a
specific numerical answer on the second. These five realised that their
answer could not be an absolute number but would have to be expressed in
some provisional way. Like the interpretation of tables (Item 1), one ex-
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
255
TABLE III
Response types for the task involving height of Sharma children (n = 14)
Number of students using it
Response type
Task (1)
Reading graphs
Task (2)
Predicting graphs
[Both tasks]
Non-response
Y
Y
Y
Non-statistical
Y
Y
Y
Partial-statistical
Y
7
Y
Statistical
14
7
[7]
planation for these differences could be a lack of emphasis in classrooms
on interpreting graphs (see Appendix 1). Since the students lacked experiences in interpreting graphs, their responses were more likely to be in
the partial-statistical categories.
Partial-Statistical Responses. Half the students were classified as using
partial-statistical approaches for the second item. When rules were applied inappropriately, non-existent patterns imposed or no patterns seen,
or exact answer to the question given, the responses were categorised as
partial-statistical (see Table I). The data revealed that one student, student 6, applied the rule for finding the mean in an inappropriate way.
When asked to predict the height of a 10-year-old from the bar graph, the
student used the add-them-all-and-divide algorithm. The student added all
the heights given in the bar graph, divided by 5 and got 84 cm! Even
further probing by the researcher did not have any positive effect on the
student’s reasoning. Misinterpretations were caused by students forcing a
pattern on the graph or not seeing the pattern. When asked to predict the
height of the 10-year-old, three students tried to force a pattern on the bar
graph. Student 14 justified this pattern in terms of a going up explanation. She said that it might be 130 cm because the first one is 100 cm and
the 8-year-old is 120 cm. The 10-year-old might be 130 cm. The trend
continues, 100, 120 and 130. The other two students cited the absence of a
pattern as the reason for their inability to predict. This occurred even in
cases where any attempt to search for a pattern made no conceptual
sense. The students believed that a pattern must exist and consequently
their inability to find the pattern resulted in their failure to offer any
prediction. For example, when asked to predict a 10-year-old’s height,
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student 20 said that he could not predict since there was no pattern on the
x-axis. During the interview, the student continued to protect his flawed
thinking rather than admit something was wrong.
I:
S20:
I:
S20:
I:
S20:
A fifth child in the family is 10 years old. Can you tell how tall the 10-year-old is?
Could be 180 cm because here [meaning 4-year-old] they are increasing by 20 cm.
So the 10-year-old is taller than the 19-year-old?
Age 10 . . . Oh I thought the fifth child. You can’t tell from the graph.
Why do you say that?
Because it goes by age 4, age 8, age 13. If it was from age 4, age 5 and age 6 you
can locate how they range.
It seems that the student believed that graphs have to show a more
definitive pattern and he was unhappy about the arrangement of categories
on the x-axis. The belief that graphs must have patterns seems to be related to other areas of the mathematics curriculum where recognition of
patterns is stressed, as well as from specific experiences with graphs in
social contexts.
Three students gave numerical answers. After being alerted that an
opinion was called for, rather than an exact mathematical response, they
did not realise that their answers could not be absolute numbers but
would have to be expressed in some provisional way. Two students
placed the 10-year-old on the x-axis halfway between the 8-year-old and
the 13-year-old and predicted the height as the corresponding point on
the y-axis as 130. The other student talked in terms of the 10-year-old
being the middle of the height of the 8-year-old and the 13-year-old,
hence 120 + 140 and divide by 2. It appears that the student tended to
worry most about What do I do? Rather than What does this information
mean? These students appear to focus only on numbers, and ignore
quantity types. For example, the height was read as 130. This emphasizes
how much but ignores of what.
It must be acknowledged that the limited use of statistical techniques
in my interviews may be a consequence of a classroom mathematics
culture that asks questions with a single answer. It takes considerable
self-confidence to say something like I think the answer might be between 120 and 140 cm or even to say You can’t tell from the graph
because that piece of information isn’t there. Furthermore, Gal (1998)
states that suggesting to students that a judgment is called for, rather than
a precise mathematical response, will make students think more about
data and not look straight away for some numbers to crunch. It appears
that this strategy might not work for students who lack experience in
explaining their answers or have strong beliefs about statistics.
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
257
Graphical Representations: A Broader Context
Interpreting Tables and Graphs. The finding that while students can
read tables and graphs, they have difficulty drawing inferences from
tables and graphs is consistent with the results reported by Bright & Friel
(1998) and Gal (1998). These researchers found that while students had
few difficulties with the literal reading of graphs, they were often unsuccessful in answering questions requiring higher order cognitive skills
such as interpreting and predicting. The findings of this study add to the
literature which reveals that while students can read bar graphs, they
experience difficulty in reading tables Y although the main influence
could be the lack of emphasis the teachers put on the reading of tables.
The use of simple summary statistics such as comparing totals and
means for small samples were used frequently in Estepa et al. study
(1999). In the current study, even after probing, very few students used
these techniques. A possible explanation for this could be that the contexts for comparing the two data sets were quite different and the
students were different ages with different statistical backgrounds, hence
the strategies employed were different. Moreover, students in my study
could have faced language difficulties. For instance, the term summarise
(Item 1) could have presented a linguistic problem as there are two
perfectly reasonable interpretations. In everyday language, it may mean
main points whereas in statistics it can be used more precisely to refer to
measures of spread and centre.
Forcing Patterns. The belief that graphs must have patterns (Item 2) is
consistent with the findings of Asp et al. (1994) and Pereira-Mendoza &
Mellor (1991). Pereira-Mendoza and Mellor found that students have a
tendency to impose patterns on data. They researched grade four and
grade six students’ understanding of the information conveyed by bar
graphs and found that errors involving pattern arrangement of the data
occurred in similar frequencies for both grades. The interview results
question the knowledge transfer theories. Students’ tendency to force
patterns may be attributed to negative transfer (Mevarech & Kramarsky,
1997). Students in my research initially learned to construct and interpret
function graphs for which recognition of patterns is stressed and they
translated these competencies to read the bar graph. It is possible that
negative transfer led some students to mistakenly conclude that all
graphs must have patterns. Another explanation is that the students are
not able to balance statistical applications and context (Friel et al., 2001).
For example, context outweighed statistical principles in the students’
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SASHI VARTA SHARMA
mind, they ignored the data and related to their personal or other school
experiences.
Rules and Procedures. Although the teacher had taught finding averages
(mean, median and mode) in a number of ways, including frequency
distribution tables and cumulative frequency graphs, the students appeared to have muddled views as to how to apply these rules. For instance, while students could work out the summary statistics involved in
simple data sets, they did not appreciate its significance or usefulness
when dealing with data tables (Item 1). Alternatively, they applied the
mean and other algorithmic procedures inappropriately to the comparing
heights questions (Item 2). The findings are consistent with the results of
Cobb (2002). Data analysis for most of the seventh graders involved
manipulating numbers in a relatively procedural way without addressing the question at hand. It appears that the limited nature of school
instruction in statistics does not provide students with opportunities to
refine their concepts and resolve ambiguities. The students learn statistics as a set of rules without learning the meaningful contexts in
which they should be applied. Teachers assume that students who learn
to process data can transfer these skills to interpreting and developing a
critical attitude to information. However, if graph comprehension is
embedded in contextual settings and the shared practical activities of
people, students need relevant experiences in which they construct and
use graphs to represent phenomena that they have somehow experienced
(Roth, 2004 p. 90).
Inconsistencies in Reasoning. A number of students could not be identified as using a consistent or pure form of reasoning but rather used a mixture of categories. The explanations given by students were not consistent
across the two items. Some students who appeared to reason according to
the statistical category on one problem seemed to use the non-statistical
category on another. Different displays seemed to induce students to use
different approaches. For example, five students were classified as nonstatistical on Item 1 (Table II). However, none of the students made such
responses on the second task (Table III). Perhaps the students already
knew the answer to the question in Item 1 (i.e., Ba is warmer). In terms
of constructivism, researchers and interviewees negotiate the interview
situation. The findings indicate that the students and the interviewer in
this study had different interpretations of the interview situation.
On the other hand, some students used different types of reasoning on
the same problem, as illustrated by a student’s response to Item 1.
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
259
Each day the temperature for Ba is greater than the temperature for Sigatoka. And
normally Sigatoka is called a valley. They are producing fruits; it rains there. My one
uncle is there. He mainly plants Chinese cabbages; because of the rain it grows so well
there.
The student’s first explanation appears to be influenced by rational
element: comparison of individual data. The second explanation reflects
a duality of statistical with non-statistical ideas that seemed to be
influenced by experience. These inconsistent responses by some students
may be accounted for in terms of constructivism. The constructivist
framework postulates that students hold mini-theories or well established beliefs when they enter instruction and they connect their new
ideas to these (Cobb, 1994; von Glasersfeld, 1993). While learning
statistics, students were connecting what was told to them in an active
way to achieve deeper understanding but often these connections led to
inconsistencies.
Prior Experiences. The finding that students base their thinking in
statistics on their prior experiences is not new. Moritz (2003) and Chick
(2000) report that the thinking of many students in their study was
dominated by past experience and this prevented them developing statistical ideas, despite being aware of a trend in the data. In some respects,
the findings of the present investigation go beyond those discussed
above. The findings demonstrate how students’ other school experiences
also influence their construction of statistical ideas. At times the inschool experiences appear to have had a negative effect on the students.
An example of negative effect that arose from other school experiences
were the students who were deeply convinced that one can only predict
from graphs that have a definitive pattern.
On the other hand, if meanings and understandings associated with
graphing are tied to the specific situations (Lave, 1991), and structured
by social situations, student opinions cannot be judged as incorrect (Gal,
1998). Perhaps, it is important to point out to students that there are
alternative points of view. For instance, a study on graphing among
scientists (Roth, 2004; Roth & Bowen, 2001) shows that when scientists
are not familiar with a graph, even if this graph is from introductory
textbooks of their own discipline, they often do not arrive at the collectively accepted standard interpretation. Some of the problems they
encountered while interpreting graphs were of the same types that have
been identified among middle school and high school students. However,
these scientists were very competent when it came to read graphs
directly related to their own discipline/research because they found
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SASHI VARTA SHARMA
them meaningful. Roth & Bowen (2001) suggested that because of
their extensive training, experience and career related success, to accept
that these scientists lack graphing skills or suffer from other deficiencies would be difficult. It appears that statistics education should
indeed avoid Fbrainwashing_ methods for alternative views are still
existent.
Interviews. Guided by the constructivist view of learning, individual
interviews were used in the study to explore and explain students’ ideas
and strategies about reading and interpreting graphs and tables. This
approach provided evidence that students often give correct answers for
incorrect reasons. For example, interpretations drawn from the data
provided were often informed by knowledge drawn from every day life.
Indeed such knowledge often replaced inferences based on the data
alone. When comparing temperatures (Item 1), four students believed
that the temperature of Ba was more than the temperature of Sigatoka. It
would be easy to conclude from the responses that the students had a
well developed concept of comparing tables. However, the justifications
provided by the students indicated that they had no statistical explanations for their responses. None of the explanations indicated any
consideration of all the information presented to the students.
In some cases students moved from inappropriate strategies to appropriate ones. This could be interpreted as another strength of interviews. During the interviews, students were asked to explain their
thought processes and thus reflect on their original and subsequent
responses. This made them think about the strategies they had used
which at times led them to modify their responses. An example was
student 3 who, when asked to find how much shorter the 4-year-old
was than the 19-year-old, initially stated 20 cm. Probing of the students’
thinking led to the student deciding that the correct response was
60 cm. However, in some cases students moved from appropriate
strategies to inappropriate ones. One of the factors that could have made
students change in this way was the students’ experiences as learners at
school. Student reasoning is rarely questioned in class, they are
questioned when they give wrong answers. It seems probable that in
my research, the students interpreted my probing as an indication that
something was wrong with their answers and so they quickly switched to
a different strategy.
Contextual Setting. The results of interview show that although contexts
may help students use prior knowledge, such situational knowledge is
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
261
diverse and can also cause misinterpretations of the information in the
data display. For instance, student 21’s personalisation of the context
brought in various interpretations of the task (Item 1) and inconsistency
in his explanations. Probably, the students’ lack of understanding of the
constraints imposed by the context distracted him from making a sensible interpretation. Janvier (1981) highlighted that the consideration of
situational knowledge increases the number of elements to which graph
readers have to attend and hence lead to a different kind of abstraction.
Given how statistics is often taught through examples drawn from Freal
life_ teachers need to exercise care in ensuring that this intended support
apparatus is not counterproductive. This is particularly important in light
of current curricula calls for pervasive use of contexts (Meyer, Dekker &
Querelle, 2001; Ministry of Education, 1992) and research showing the
effects of contexts on students’ ability to solve open ended tasks (Cooper
and Dunne, 1999; Sullivan, Zevenbergen & Mousley, 2002). For instance, the study by Cooper and Dunne (across a whole range of school
mathematics tasks) found that some pupils have a greater facility in
recognising whether they are being asked to play a Fschool maths_ game
or an Feveryday life_ game.
Conversely, in spite of the importance of relating classroom mathematics to the real world, the results of my research indicate that students
frequently fail to connect the mathematics they learn at school with
situations in which it is needed. For instance, while students could
calculate summary statistics from data sets, they had difficulty summarising data presented in table (Item 1). Clearly, the results support
claims made by Lave (1991) that learning for students is situation
specific and that that connecting students’ everyday contexts to academic
mathematics is not easy. One reason for this could be that when
everyday experience and knowledge of the student are connected to
school mathematics during the process of contextualizing, a new set of
demands is created that requires the students to identify this process of
recontextualisation (Dapueto & Parenti, 1999).
LIMITATIONS
The findings reveal that many of the students used beliefs and personal
and social experiences to explain their thinking. It must be acknowledged that the open-ended nature of the tasks and the lack of guidance
given to students regarding what was required of them certainly influenced how students explained their understanding. The students may
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not have been particularly interested in these types of questions as they
are not used to having to describe their reasoning in the classroom. Some
students in this sample clearly had difficulty explaining explicitly about
their thinking. Students who realised that Ba is warmer than Sigatoka
(Item 1A) had a difficult time articulating exactly how data could be
used to make a sensible interpretation in such situations. The issues of
language use are particularly more important for these students, the
students face schooling in a second language that is not spoken at home.
Another reason could be that such questions do not appear in external
examinations. Instruction in Fiji centers mostly around presentation of
short answers of the correct/incorrect type to procedural questions. Although the study provides some valuable insights into the kind of thinking
that high school students use, the conclusions cannot claim generality
because of a small sample. Additionally, the study was qualitative in
emphasis and the results rely heavily on my skills to collect information
from students. Some implications for future research are implied by the
limitations of this study.
IMPLICATIONS
FOR
FURTHER RESEARCH
One direction for further research could be to replicate the present study
with more carefully designed research interview and include a larger
sample of students from different backgrounds so that conclusions can be
generalised.
Secondly, this small scale investigation into identifying and describing students’ reasoning has opened up possibilities to do further research
at a macro-level on students’ thinking and to develop more explicit
rubric for each category of the framework. Such research would validate
the framework of response types described in the current study and raise
more awareness of the types of thinking that need to be considered when
planning instruction and developing students’ statistical thinking.
Thirdly, if context is important for graph interpretation, then one
needs to consider several elements when designing tasks. First of all,
researchers cannot make appropriate assessments without also having
some knowledge about the range of embodied experiences in the real
world of the learner need to choose tasks where student context knowledge base is good. The interview results show that personalisation of the
context can bring in multiple interpretations of tasks and possibly different kinds of conclusions. At this point it is not clear how a learner’s
understanding of the context contributes to his/her interpretation of data
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
263
represented in tables and graphs. There is a need to include more items
using different contexts in order to explore students’ conceptions of
graphs and related contexts in much more depth.
While some pupils answered part (1) of Item 1 with reference to the
numerical values in the table, others used their prior knowledge about the
two towns. It would be interesting to see how easily (or whether) students who argued on the basis of prior knowledge on this question could
be persuaded to argue purely on the basis of the numerical data. Future
research could incorporate this into the interview procedure to explore
this issue in more depth.
The picture of students’ thinking in regards to graphical representations is somehow limited because students responded to only one item
related to tables and one item related to bar graph. There is a need to
include more items using different social contexts in order to explore
students’ conceptions of tables and bar graphs in much more depth.
Additionally, study of histograms and pie charts are considered important at the upper secondary levels (see Appendix 1). Histograms are
often used to organise continuous data whereas a pie chart may be
used to efficiently compare data. It would be interesting to explore how
changing the context of the representation influences the types of responses exhibited by the students.
Researchers can accurately assess their subjects’ understanding
through individual interviews. The interview results provide evidence
that students often experience difficulty when speaking about tables.
However, in the present investigation I overcame these difficulties by
restating a task or changing the wording. For instance, some students
were not familiar with the word Fsummarise_ in Item 1 and an example
of summary statistics was used to clarify the situation. This would have
not been possible in a written survey.
Another implication relates to culture. Unlike Estepa et al. study
(1999), very few students in my study used statistical techniques on Item
1. One explanation for this could be the cultural context. Metz (1997)
suggests that to adequately understand students’ cognitive constructions
and beliefs, we need to consider the culture in which students participate.
Watson & Callingham (2003) note that students in Fother cultural
settings_ may respond differently to their Australian counterparts,
particularly to context-based items used in their studies. It would be
interesting to determine how cultural practices influence conceptions of
graph comprehension. Such an investigation would involve documenting
cultural practices (particularly those involving statistical representations)
in which students typically participate in Fiji and in western culture,
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SASHI VARTA SHARMA
delineating the differences, and relating these contrasts to differences in
the forms of reasoning that students in the different cultures have
developed.
Finally, the place of statistics has changed in the revised mathematics
prescription. Statistics appears for the first time at all grade levels (Fijian
Ministry of Education, Women, Culture, Science and Technology, 1994).
Like the secondary school students, primary school students are likely to
resort to non-statistical or deterministic explanations. Research efforts at
this level are crucial in order to inform teachers, teacher educators and
curriculum writers.
CONCLUDING THOUGHTS
Although subjective types of thinking permeate our students’ lives,
statistical curricula all over the world presumes the Fcorrectness_ of the
mathematical model and ignores beliefs and experiences that do exist in
real life situations. This presents a real dilemma which needs to be
resolved if statistics education is to flourish. For example, if students
come to the class with the view that all graphs have patterns and the
teacher is trying to teach the key concepts related to the use and
interpretation graphs, then how this can be done in a way that does not
denigrate the first view needs to be investigated. It is not adequate to
consider this just as a Fbias,_ the students, after all, require the pattern
view in other learning areas. Perhaps, it is important to point out to
students that there are alternative points of view. It appears that statistics
education should indeed avoid Fbrainwashing_ methods for alternative
views are still existent (Roth & Bowen, 2001). It is hoped that the
findings of this study will generate more interest in research with respect
to subjective ideas that students possess and relevant contexts. Teachers,
curriculum developers and researchers need to work together to find
better ways to help students interpret tables and graphs.
APPENDIX 1 MATHEMATICS EDUCATION
IN
FIJIAN SCHOOLS
Background
Primary schools teach classes 1 to 8 (in some cases classes 1 to 6)
whereas secondary schools teach Forms III to VII, and Junior secondary
HIGH SCHOOL STUDENTS INTERPRETING TABLES AND GRAPHS
265
schools teach Forms I to IV. The following external examinations involving mathematics are taken by pupils while at school:
Class 6
Form II
Form IV
Form VI
Form VII
Fiji Intermediate Examination in which mathematics is
compulsory.
Fiji Secondary Schools’ Entrance Examination in which
mathematics is compulsory
Fiji Junior Certificate Examination in which mathematics is
compulsory
Fiji School Leaving Certificate Examination in which
mathematics is not compulsory
Fiji Seventh Form Certificate Examination in which mathematics is not compulsory
Statistics
Statistics is taught as one of the topics in the mathematics syllabus and is
first introduced in Form 11 (Class 8) where the average age of a pupil is
13 years. Statistics is then taught in bits and pieces right up to Form VII.
A brief summary of statistics taught at different levels is given below:
Form II (Class 8). Collecting information, representing information,
interpreting representations, average (mean), pictograms, frequency
tables, bar graphs, pie charts.
Form III. Representing statistical data in graphical or chart form,
computing sample mean, mode, median and range, organising data from
a sample into a frequency distribution and representing this by a
frequency line graph or a histogram.
Form IV. Computing statistics: mean, median, upper and lower quartiles,
interquartile range. Representing data: frequency polygon, cumulative frequency table, cumulative frequency graph. Elementary ideas in probability.
Form V. Classification of data, statistical graphs, including frequency
and cumulative frequency curves, median and mean as measures of
central tendency, ideas of spread, including standard deviation, simple
probability and relative frequency.
Form VI. Probability: sample space, mutually exclusive events, independent events. Populations and samples: mean, median, standard
deviation and range as examples of population parameters, samples, ran-
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dom samples, frequency distributions, sample statistics. Distributions: the
binomial; distribution taken as an example of a discrete distribution, mean
of binomial distribution. The normal distribution as an example of a
continuous distribution, z score.
Form VII. Probability, Statistics and Computing
Choose only one of the options
Option A: Probability and Statistics
Option B: Probability and Computing
Option C: Statistics and Computing
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Mathematics, Science and Technology Education,
The University of Waikato,
62 May Street, Hamilton, NA, New Zealand
E-mail: sashi@waikato.ac.nz