Original Article
Hold my calls: an activity for introducing the
statistical process
Todd Abel
Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina, USA
e-mail: abelta@appstate.edu
Lisa Poling
Reich College of Education, Appalachian State University, Boone, North Carolina, USA
e-mail: polingll@appstate.edu
Summary
Working with practicing teachers, this article demonstrates, through the facilitation of a
statistical activity, how to introduce and investigate the unique qualities of the statistical
process including: formulate a question, collect data, analyze data, and interpret data.
Keywords:
Teaching; Statistical literacy; Problem solving cycle; Statistical process; Teaching
statistics; Data investigation.
INTRODUCTION
Statistical reasoning defined by Gal (2000, 2002)
is the ability to problem solve and reason with data
in a way that makes data succinct and understandable. To be able to reason statistically, statisticians and statistical educators have developed
standard phases of the statistical process that
can be described as follows: (1) specify the problem and outline a plan; (2) collect the required
data; (3) analyse and represent the data; and
(4) interpret and discuss the findings (Marriott
et al. 2009; Guidelines for Assessment and
Instruction in Statistics Education (GAISE), 2005;
Wild and Pfannkuch 1999).
Teachers of statistics should understand the
statistical process not only as a way of doing statistics but as a primary means of motivating, introducing, investigating and applying statistical
concepts. Indeed, to be engaged in the conceptual
understanding of statistics, both learner and
instructor must value and understand the process
of statistical thinking. Unfortunately, ‘inappropriate reasoning about statistical ideas is widespread and persistent, similar at all age levels,
and quite difficult to change’ (Garfield and BenZvi 2007, p. 374).
The purpose of this article is to describe a statistical activity used with practicing teachers to introduce and investigate the statistical process.
While the activity is intended for secondary level
learners and could be used to teach particular
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statistical content, the primary intent is to highlight and discuss the statistical process.
The statistical process
Learning experiences are bound by how individuals navigate information and what they deem to
be pertinent and essential. According to Wild and
Pfannkuch (1999), the ‘ultimate goal of statistical
investigation is learning in the context sphere’
(p 225). When selecting tasks for the classroom,
the significance of the activity to the individual
and the focus on statistical reasoning skills
become paramount to the learning experience.
Designing an investigation based on data requires
that teachers identify issues of interest and frame
the questions, ‘together with considerations of
practicalities in collecting the data’ (MacGillivray
2007, p. 53), in a manner that adheres to the
most appropriate measure of statistical analysis
(MacGillivray 2007).
Using a statistical process model provides a
framework that allows individuals to proceed
through an exploration that promotes uniformity
and reliability. The problem-solving approach in
teaching statistics is ‘considered to improve
students’ skills, particularly as they interact with
real data’ (Marriott et al. 2009, p. 3). Although
various models of the problem-solving approach
exist, for this study, we used a four-step model,
the GAISE framework (2005). Designed by statisticians and statistics educators in the USA, the
© 2015 The Authors
Teaching Statistics © 2015 Teaching Statistics Trust, 37, 3, pp 96–103
Hold my calls
GAISE framework provides a conceptual framework for K–12 statistics education. The GAISE
framework outlines a statistical process consisting of four components: (1) formulate questions;
(2) collect data; (3) analyse data; and (4) interpret results (GAISE, 2005). The GAISE framework is then organized into three sequential
developmental levels. As students progress
across levels, learning becomes more student
directed and less teacher directed, and student
engagement becomes more nuanced and sophisticated (figure 1).
Importance for teachers
Ben-Zvi and Garfield (2004) identify various factors
that challenge both learners and teachers of statistics: statistical ideas are complex, lack of mathematical knowledge impedes the computational
aspect of statistics, the context in which statistical
problems are presented is confusing and
individuals transfer limited understanding of
mathematics to statistics focusing on getting
the correct answer but not interpreting results.
To create environments that scaffold learning
and engage students, instructional strategies
must provide support so that students can experience the full range of the statistical process: enter the problem, allow for dissonance
and make connections to existing concepts
(Garfield and Ben-Zvi 2009). Students, as constructors of knowledge, must be actively
engaged as they explore probability and statistics. Teachers therefore must provide consistent feedback, giving students opportunities
to practice using statistical techniques and
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address misconceptions as sources of knowledge (Garfield 1995; Garfield and Ben-Zvi
2007).
Just as students need experience to understand
a concept, instructors need to first experience a
concept in order to fully understand and more
effectively teach it (Polly and Hannafin 2011).
Thus, in order to help teachers experience and
understand the statistical process, we developed
and implemented a task during a professional development workshop for grades 6–12 mathematics
teachers that used the GAISE framework to engage
participants in that process. The underpinning and
consistency of using the GAISE framework allow
learners to organize and enter into a statistical
problem, providing the scaffold that learners need
in order to make sense of specific statistical content. Creating such carefully designed sequences
helps students improve reasoning and understanding (Garfield and Ben-Zvi 2007). Moreover,
it emphasizes the nature of statistics and the
practices and processes involved in statistical
thought.
CONTEXT
In subsequent sections, an activity is outlined
that was used with practicing teacher, showing
the four steps in the problem-solving approach
to teaching statistics. Four different groups of
practicing teachers engaged in the activity at
various times during 2 days of professional development. As a result, the data displayed in the
examples differ depending on the group.
Fig. 1. Guidelines for Assessment and Instruction in Statistics Education framework
© 2015 The Authors
Teaching Statistics © 2015 Teaching Statistics Trust, 37, 3, pp 96–103
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Todd Abel and Lisa Poling
INTRODUCING THE STATISTICAL PROCESS
WITH AN ACTIVITY
Adapting the Cell Phone Impairment? activity
from the Statistics Education Web (STEW) (nd)
of the American Statistics Association, the lesson
investigated the relationship between cell phone
usage and reaction time. Changes in the activity
were made to conform to the particular context
and emphasize the statistical process, permitting
multiple levels of engagement.
Formulate questions
The first phase of the statistics process involves
formulating questions to be investigated. Participants were given the following prompt:
According to the National Safety Council, cell
phones are a factor in 1.3 million traffic accidents each year, resulting in thousands of
deaths and injuries. Many have conjectured
that this is due to decreased reaction time
and attentiveness. What questions does this
prompt you to ask?
The open-ended nature of the prompt was
useful for eliciting a variety of possible responses. This highlighted the questioning
habits of teachers in the groups and motivated
a discussion of characteristics of good statistics
questions.
Initial questions tended to either address the
reliability of the statement or propose possible
contributing variables. For instance,
• how do they know cell phones are a factor?
• where did the number come from – define traffic
accident?
• how is cell phone use reported?
• how does this compare to other distracting activities as a factor?
• is it just cell phones or any distraction?
• what are the demographics of drivers?
• how will you measure reaction time?
• hands-free versus not?
• does type of phone matter?
• how is phone being use?
• does age matter?
• were there other possible reasons? Is cell phone
the only/leading factor?
• men or women?
• does type of driving (setting) make difference?
These questions lack specificity and do not anticipate variability in a way that can be systematically investigated. Many of them are, however,
important questions to consider and were useful
for considering sources of variation and possible
avenues for investigation. Listing such questions
encouraged the groups to consider the characteristics of good statistical questions and prompted
a discussion about what could or should be
investigated. In table groups, participants then
proposed possible statistics questions refined
from the initial list. A direct investigation of
the reasonableness of the statistics from the
prompt was unrealistic, so the focus was placed
on the conjectured relationship between cell
phone usage and reaction time. Ultimately,
each group settled on the following question:
Does cell phone usage have an impact on reaction time? Although it does not consider any
effects on driving, the final question could guide
a classroom investigation and was compelling
to the participants.
The nature of the initial questions highlights
that teachers, as with students, need opportunities to practice and develop the statistical
habit of mind of questioning. In the classroom,
as in our example, it may be necessary for
instructors to use questioning to guide students to a particular well-defined statistical
question. If the goal of the lesson is to understand the statistical process, however, it is
important that students have opportunities to
propose questions, and that viable alternatives
are recognized.
Collect data
Data collection procedures must account for potential sources of variation in data. While students
need opportunities to design data collection procedures, allowing a class to design an experiment
from scratch is often impractical. Implementing
ideal data collection procedures can also be difficult. In many cases, a classroom pilot experiment
may highlight the necessary content while permitting students to be actively involved in data
collection. In this activity, basic data collection
procedures were given, but participants were
asked to analyse the procedures for possible
sources of variability and then specify ways to
minimize them.
The following data collection procedures were
proposed to the group:
1. Participants worked with a partner, and two sets
of partners formed a table group of four.
2. Reaction time was quantified using a ruler drop
method. One partner (the dropper) held a ruler
and dropped it at random, while the other partner (the catcher) held their hand at the bottom
end of the ruler and caught it as it fell. Reaction
© 2015 The Authors
Teaching Statistics © 2015 Teaching Statistics Trust, 37, 3, pp 96–103
Hold my calls
99
Table 1. Types of analysis
Individualized
Aggregate
Separate conditions
Difference
Type I–C: graph compares each individual’s reaction
time for each condition: with and without a cell phone.
Type A–C: graph compares the aggregate data for each
condition: with and without a cell phone.
Type I–D: graph illustrates the difference in two
reaction times for each individual.
Type A–D: graph illustrates the difference in reaction
time for the overall data set, instead of by individual.
time was measured by recording the distance the
ruler travelled before being caught.1
3. A coin flip determined whether the table group
would measure reaction time with a cell phone
or without a cell phone first.
a. Each partner took a turn as catcher and as
dropper without a cell phone.2
b. Each partner took a turn as catcher and as dropper. During a particular turn, the two catchers
in a table group talked to each other on cell
phones. To simulate a conversation, each participant created a list of ten words, which they
read to the other catcher, who had to try and
repeat as many back as they could. While doing
that, the dropper would drop the ruler.
4. Each table group input their data into a collaborative spreadsheet to make it available to the
whole group.
When prompted, participants identified and
clarified several other possible sources of variability, such as the following:
• How will the dropper hold the ruler? Where and
how should the catcher hold their hand?
• Where should the measurement be taken? What
finger will locate the measurement?
• What level of precision is expected in the length
measurements (nearest inch, quarter inch, etc.)?
• How many trials with and without the cell phone?
1
Note that the measurement of reaction time using
these data collection procedures is actually a length
measurement. Because the ruler will accelerate due
to gravity, the length and time are not proportional.
During data analysis, one could convert the lengths
2
to times using the equation d = 192t , where t is time
in seconds and d is distance in inches. However, the
question can be addressed without attending to that
conversion, as it was in the instance presented here.
Participants did acknowledge the length-to-time conversion issue, and one group modelled the conversion.
The results were close enough to proportional, and the
times short enough, that this group decided to ignore
the conversion to time and work strictly with the recorded lengths. Thus, in what follows, reaction time
refers to the point on the ruler where the ruler was
caught.
2
If available, reaction time apps are freely available for
laptops, tablets and smartphones and could be used in
place of the ruler drop method.
© 2015 The Authors
Teaching Statistics © 2015 Teaching Statistics Trust, 37, 3, pp 96–103
Using a think, pair, share protocol, data collection procedures were standardized as a group.
Participants were also asked to identify limitations
of the data collection plan, such as the limited population (limited to the classroom), the minimal
number of trials and possible confounding of data
from subjects being observed by other subjects.
Despite the quasi-experimental nature of data
collection, these procedures were sufficient to highlight key characteristics of experimental design and
illustrate the nature of the data collection phase of
the statistical process. Issues of data entry, handling and preparation were not emphasized here.
Analyse data
During the data analysis phase of the statistical
process, data are explored and represented. In instructional settings, the data analysis phase may
serve as an opportunity to motivate and implement new techniques, to further explore or apply
previously developed techniques, or to compare
different analyses. The goal of this particular activity was to introduce and investigate the statistical process rather than any particular content, so
each table group was asked to examine the data
by creating a visual representation of their choice,
an important first step in data analysis. This permitted engagement at different levels, thereby
creating opportunities to discuss the statistical
process and differentiate levels of understanding.
Although this group did not explore further, more
sophisticated analysis techniques would certainly
be appropriate given different learning goals.
It was observed that table groups organized their
choice of visual representation by two main categorizations: individualized versus aggregate and separate versus difference as shown in table 1. Below,
examples of each type are shown, and conceptual
underpinnings (or misunderstandings) underlying
each are discussed. Note that each table group of
four participants created a graph for the data from
their entire class. Four different classes took part
in the activity, so the data differ somewhat among
the graphs shown below.
Type I–C table groups placed an emphasis on
representing each individual time, both with and
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Todd Abel and Lisa Poling
Fig. 2. Examples of type I–C graphs
without the cell phone. These graphs attempted to
give an overall sense of which time was generally
higher without regard to the magnitude of the differences for each individual or attempt to summarize the data. A variety of type I–C representations
was used. Some table groups created double bar
graphs showing each individual’s reaction time
with and without the cell phone. Other table
groups created a scatterplot, with each ordered
pair (without cell phone and with cell phone)
representing an individual. Effect on reaction time
was visualized by comparing points to the line
y = x. Other table groups created line graphs that
attempted to show the same person-by-person
comparison. While each of these graphs gives the
overall impression that the cell phone reaction
times were typically longer, they do not give any
specific information about the group as a whole,
attending instead to individuals. In addition, the
line graph is particularly misleading since the data
on the horizontal axis (individuals) are not ordered
(figure 2).
Only one table group created a graph of type
I–D. This group calculated the change in reaction
time (with cell phone) and (no cell phone) for
each individual, then plotted each one using a
type of bar graph. This does attend to the differences in reaction time, but still emphasizes each
Fig. 3. An example of a type I–D bar graph
individual rather than giving any sense of the data
as a whole. The graph itself is problematic – it is
not initially clear that each bar represents an individual (on first glance, it looks more like a histogram) and makes the horizontal axis appear to
be ordered when it is not (figure 3).
The most popular type of graph was type A–C.
These groups compared the aggregated data
with and without a cell phone and used summary statistics to display information about each
treatment on the entire group. One table group
simply found the mean of the reaction times
with and without the cell phone and created a
bar graph to show the two means. More common, however, was a comparative box plot –
one box plot for the aggregate data within each
condition. Such graphs are moving beyond
looking at each individual subject and instead
taking advantage of aggregate data. However,
they are not taking account of the experimental
situation and hence not conveying any specific
information about the differences. This problem
is emphasized by an incorrect interpretation described below (figure 4).
Type A–D graphs considered the difference (with
cell phone and without cell phone) for each individual then created graphs that looked at the data
as a whole instead of individually. For instance,
one group created a box plot using the differences,
and several groups created histogram plots for the
set of data of the differences. Such graphs illustrated the most sophisticated representation of
the data, moving beyond individual responses and
giving a sense of the typical difference in reaction
times. Note how only the A–D graphs can pick up
both the overall effects and the nuances of effects
and variation in individuals of using cell phones
(figure 5).
Allowing participants to choose and create their
own displays of data provided opportunities to
compare the different data displays, highlighting
the strengths and weakness, and possible misapplications, of each. It also highlighted that the
choice of appropriate graphs is an important topic
© 2015 The Authors
Teaching Statistics © 2015 Teaching Statistics Trust, 37, 3, pp 96–103
Hold my calls
101
Fig. 4. Examples of type A–C graphs
Fig. 5. Examples of type A–D graphs
for teachers to consider, and one with which they
may need to gain experience. Comparing different graphs also prompted discussions about what
attributes of the data were important for addressing the original question.
• ‘[The] median is higher for use of the cell phone.
Maximum data is much higher for reaction time
with cell phone.’
• ‘All measurements of quartiles without the cell
phone are less than the quartile measures of with
a cell phone.’
Interpret results
Asking participants to draw and support conclusions from their graphs also highlighted
some misconceptions and problems with choice
of graphical representation. For instance, the
table group shown below created a comparative box plot and concluded that the graph
‘shows that over 75% can react faster without
the distraction of a cell phone!’
While it is true that 75% of the cell phone
reaction times were greater than at least
75% of the without cell phone reaction times,
this does not imply that 75% of individuals
had faster reaction times without cell phones.
Since differences on individuals were not used,
the graph cannot be used to make statements
(figure 6).
To emphasize the importance of interpreting
results in a meaningful and well-supported way,
participants were asked if they were confident
enough in their conclusions to
It can be challenging to generalize from data without making unsupported claims. Each table group
was asked to write the conclusions that they could
draw from their graphs. Several examples are represented in the pictures above. The participants’
conclusions often reflected the type of representations and the sophistication of their understanding of data. For instance, a very general conclusion
such as the reaction time is greater when using a
cell phone reflected the nonspecific graphical
analysis undertaken in the task. Type I–C and type
I–D graphs did not illustrate any specific measurements of differences in reaction time, permitting
only general overall impressions of the effect.
Table groups creating type A–C and type A–D
graphs, however, could use summary statistics to
support their conclusions. Although type A–C
graphs ignored the paired nature of the data, in
this particular case, the contrast between using
and not using cell phones was far greater than variation across individuals, allowing participants
who created comparative box plots to make statements such as the following:
© 2015 The Authors
Teaching Statistics © 2015 Teaching Statistics Trust, 37, 3, pp 96–103
1. change their own behaviour,
2. use it as evidence to influence the behaviour of
others or
3. present it to other statistically proficient people.
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Fig. 6. A misleading representation of data with misconceptions in the conclusion
The subsequent discussion allowed the group to
not just propose conclusions but also support them
and analyse the strength of those conclusions.
DISCUSSION
Summarizing the activity
Since the learning goal of this activity was the statistical process, the groups identified the portion
of the activity that corresponded to each phase
of the statistical process, identifying the key characteristics. For instance, standardizing data collection procedures in order to isolate a single
source of variability (as much as possible) was
noted as a key characteristic of the collect data
phase. Making choices about how to organize,
summarize and display the data was identified
as key characteristics of the analyse data phase.
The reaction time task is typical of many activities in that it permits adaptations that would increase or decrease the statistical sophistication of
any given component. For instance, the question
could be posed for the class, and only individualized data might be considered. Such adaptations
may be appropriate for an initial exposure to the
statistical process if the issues of variability involved in each component are discussed. For a
more sophisticated approach to the activity, students could design data collection procedures that
incorporate random sampling outside the classroom, and data analysis could include tests of significance and measures of variability.
CONCLUSION
To promote the statistical process, the implementation of the problem-solving approach for statistics,
Todd Abel and Lisa Poling
teacher participants became students. Participants
were asked to identify the key ideas they would like
to carry forward into their classrooms. All groups
identified the nature of statistical investigation as
a major point of emphasis. For instance, one group
noted that they realized they needed to ‘let students be flexible – [it’s] not a linear path’, while another described it as ‘not a set pattern’. The idea of
different levels of sophistication was also meaningful to many groups.
Opportunities such as this are important for
both teachers and students. Using the statistical
investigation process as a framework for the activity structured the experience and supported
participant growth in statistical understanding.
This activity illustrated how intentionally designed
learning opportunities can support understanding
of the statistical process and the nature of statistical work.
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