Hybrid Tasks Promoting Statistical Thinking And Critical Thinking Through The Same Mathematical Activities

Aizikovitsh-Udi, E., Clarke, D., & Kuntze, S. (2014). Hybrid tasks: Promoting statistical
thinking and critical thinking through the sema mathematical activities. PNA, 8(3), 115-126.
HYBRID TASKS: PROMOTING STATISTICAL
THINKING AND CRITICAL THINKING
THROUGH THE SAME MATHEMATICAL
ACTIVITIES
Einav Aizikovitsh-Udi, David Clarke, and Sebastian Kuntze
Even though statistical thinking and critical thinking appear to have
strong links from a theoretical point of view, empirical research into the
intersections and potential interrelatedness of these aspects of
competence is scarce. Our research suggests that thinking skills in both
areas may be interdependent. Given this interconnection, it should be
possible to stimulate both forms of thinking through the one task. This
paper explores the implications of an exploratory qualitative study into
processes when working on tasks encompassing both these areas for the
design of tasks that simultaneously stimulate critical thinking and
domain-specific thinking.
Keywords: Critical thinking; Hybrid tasks; Statistical thinking; Task design
Tareas híbridas: promover el pensamiento estadístico y el pensamiento
crítico por medio de las mismas actividades matemáticas
Aunque desde un punto de vista teórico, el pensamiento estadístico y el
pensamiento crítico parecen tener fuertes conexiones, la investigación
empírica sobre la intersección y la potencial interrelación entre estos
aspectos de competencia es escasa. Nuestra investigación sugiere que
las habilidades de pensamiento en ambas áreas pueden ser
interdependientes. Dada esta interconexión, debería ser posible
estimular ambas formas de pensamiento mediante una tarea. Este
artículo explora las implicaciones de un estudio cualitativo exploratorio
sobre los procesos de pensamiento en tareas que abordan ambas áreas
para el diseño de tareas que simultáneamente estimulan el pensamiento
crítico y pensamiento específico de un dominio.
Términos clave: Diseño de tareas; Pensamiento crítico; Pensamiento estadístico;
Tareas híbridas

116 E. Aizikovitsh-Udi, D. Clarke, and S. Kuntze
PNA 8(3)
Mathematics classroom instruction is generally organized around and delivered
through students’ activities on mathematical tasks (Doyle, 1988). Notably, in all
of the seven countries that participated in the TIMSS 1999 Video Study, eighth-
grade mathematics was most commonly taught by spending at least 80% of
lesson time in mathematics classrooms working on mathematical tasks
(Gallimore, Garnier, Givvin, Hollingsworth, Jacobs, Chiu, et al., 2003). For our
purposes, complete description of a mathematical task requires specification of
the intentions, actions and interpretations of both teacher and student/s, together
with details of the context in which the task was undertaken and by whom
(Mesiti & Clarke, 2010). In this paper, task mostly refers to the written stimulus,
although we also provide the details required to understand the performative
realisation of the task as a mathematical activity. Classroom activities are
coherent actions shaped by the instructional context, in general, and, in
particular, by what is taught through the use of tasks (Stodolsky, 1988). The tasks
that teachers assign can determine how students come to understand what is
taught. In other words, tasks serve as a context for students’ thinking, during and
after instruction. Doyle (1983) argues the point that “tasks influence learners by
directing their attention to particular aspects of content and by specifying ways of
processing information” (p. 161).
To achieve quality mathematics instruction, the role of mathematical tasks to
stimulate students’ cognitive processes is crucial (Hiebert & Wearne, 1993).
Contemporary curricula prioritise more than just procedural knowledge. In the
mathematics curricula of countries as culturally and geographically distant as
China, Australia and Finland, student thinking is promoted in more sophisticated
terms than simply knowledge of facts and procedures. Korea and Singapore, in
particular, seek to retain the high level of mathematical competence documented
in TIMSS and PISA results, while improving higher order thinking and problem
solving expertise. This poses the question as to whether both can be achieved.
The possibility is explored in this paper that the same tasks might be employed to
realise both goals. The tasks employed came from the research of Kuntze and his
colleagues, the actual interviews were conducted by Aizikovitsh-Udi, the
analyses of the task responses with respect to statistical thinking and critical
thinking were coordinated by Kuntze and Aizikovitsh-Udi and the task-specific
interpretive analysis was done by Clarke.
In summary, the centrality of tasks in mathematics classroom is evident from
theoretical perspectives as well as in empirical results from international
comparative studies. The role of mathematical tasks provides a key to any
attempt to understand teaching and learning in research on classroom practices in
mathematics. But can a given task stimulate and promote both discipline-specific
thinking and more generic forms of higher-order thinking? To explore this
question, we take Statistical Thinking (ST) and Critical Thinking (CT) as our
exemplars of the two modes of thought.

Hybrid tasks: Promoting statistical thinking and… 117
PNA 8(3)
STATISTICAL THINKING AND CRITICAL THINKING
In a well-known definition of statistical literacy by Gal (2004), a “critical stance”
is included among the key attitudes for successful CT-hence, Gal includes such
attitudes in his definition of statistical literacy (Reading, 2002; Wallman, 1993;
Watson, 1997). However, being critical in statistical contexts is not only an
attitude. It is possible to describe specific abilities that have to be used in order to
critically evaluate statistical data. Two key concepts or overarching ideas in CT
relevant for a critical evaluation of data are manipulation of data by reduction
(Kröpfl, Peschek, & Schneider, 2000) and dealing with statistical variation (e.g.,
Watson & Callingham, 2003). Successfully manipulating data by reduction
requires the awareness of such things as that calculating a mean value affords an
overview on the original data, but it reduces the initial information. Hence, the
resultant statistical value is (only) an indicator corresponding to a specific
mathematical model, and we should not forget that it reflects only a part of the
information. In order to critically evaluate the data, we might need additional
information about the distribution, such as the variance, or information about
extreme values.
CT skills rely on self-regulation of the thinking processes, construction of
meaning, and detection of patterns in supposedly disorganized structures (Ennis,
1989). CT tends to be complex and often terminates in multiple solutions, each
with advantages and disadvantages, rather than a single clear solution. It requires
the use of multiple, sometimes mutually contradictory criteria, and frequently
concludes with uncertainty. This description of CT already suggests links with
ST, such as dealing with uncertainty, contradictions and a critical evaluation of
given claims (McPeck, 1981). Dealing critically with informationa crucial
aspect for both domainsdemands critical/evaluative thinking based on rational
thinking processes and decisions (Aizikovitsh-Udi, 2012; Aizikovitsh-Udi &
Amit, 2008). Can a single task be used to elicit and promote both forms of
thinking?
In-depth analyses are required into how CT and ST may interdepend. In
order to design the type of hybrid tasks proposed here, it is essential that we
understand the connection between the two forms of thinking that provide the
specific goal for the use of such tasks in mathematics classrooms and how these
connect to task characteristics.
INVESTIGATING HYBRID TASKS
In attempting to stimulate particular thinking skills, it might be thought best to
target either discipline-specific thinking (e.g., twelve numbers have a mean of 10
and a standard deviation of 2, what might the numbers be?) or generic CT skills
in discipline-free contexts (e.g., five people are isolated by flood, under what

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circumstances would it not be appropriate to share the available food equally?).
Hybrid tasks seek to promote both. This paper explores the actions such tasks
promote, the rationale for their use, and their design characteristics.
In order to explore thinking processes related to tasks in the domains of both
ST and CT, individual semi-structured interviews were conducted with mathe-
matics teachers. By using mathematics teachers as subjects, basic content compe-
tence can be assumed and it becomes possible to examine their content-related
higher order thinking skills, both in terms of CT and CT. The interviews focused
on thinking-aloud when solving tasks and each lasted about 40–50 minutes.
Beyond solving the tasks, the interviewees were also free to give their personal
views on the tasks.
In the following section, the results from one interview with a single teacher
are used to exemplify the sort of data generated and the type of task likely to
stimulate ST and CT. The analysis concentrated on identifying ST and CT as
employed by the interviewees. Our interest is not just in the capacity of a single
task to stimulate both ST and CT, but whether the interaction between ST and
CT made them mutually supportive or disruptive. A first analysis was done
focusing on ST only (Watson, 1997), then a second analysis employed a CT
point of view (Ennis & Millman, 2005). We then carried out a combined
interpretative analysis in order to examine relationships between CT and ST
elements. In this methodological approach, the analyses were done by two coders
working in parallel. Figure 1 shows the first task analysed.
Figure 1. Task “population” (Kuntze, Lindmeier, & Reiss, 2008)

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Nena was an experienced US secondary mathematics teacher with 20 years of
mathematics teaching experience. She was dedicated to improving her teaching
and had been participating in a year-long professional development program for
mathematics teachers at the time this interview occurred. In the interview, Nena
was asked to solve the problem in Figure 1, while thinking aloud.
Interviewer: What do you think? From which year on has the population been
decreasing?
Nena: Well…the population has been decreasing since 1963 until about 1973,
where the population begins to rise a bit...but it is still down since 1963...
Interviewer: Are you confident about it?
Nena: Of course… I am sure! You can look at this problem mathematically,
anywhere there is a positive slope you could say the population is
increasing, where there is a negative slope the population is decreasing.
But compared to 1963, it’s always been down.
Dealing with assumptions is one of the key elements of CT (Ennis & Millham,
2005). Nena initially assumed that the graph of births completely determined the
population development. Even when asked to reflect on this assumption, Nena
did not generate possible counter-arguments for testing her initial assumption,
nor did she appear to question this assumption. She tended to seek confirming
evidence rather than evidence that might challenge her initial assumption.
Seen from the perspective of ST, Nena chose an inappropriate statistical
model for interpreting the data given in the diagram (description of this task in
Kuntze et al., 2008). She appeared to focus on the data related to the births only,
and she deduced her conclusions from a mathematical consideration of slopes.
Even when encouraged by the interviewer, she did not check this model against
the full data given in the diagram.
Nena’s answers show deficits both in CT and ST. Looking at the
relationships between CT and ST: At the very beginning, Nena shows only a
partial perception of the evidence, focusing on the birth data from 1963 on. This
selective focus may have been a result of the headline given in the diagram (“The
Germans don’t have enough children”). This headline may have triggered Nena’s
misinterpretation of the births as determining the population, from the turning
point in 1963. It is interesting that Nena emphasised that it was possible to “look
at this problem mathematically”, which suggests that she saw a discrepancy
between looking at the situation from the perspective of a mathematical model
and looking at it from the perspective of the context (population and children
born in Germany). Possibly the mathematical or statistical model is taken as an
authority that is used to justify the appropriateness of the assumption instead of
questioning the model chosen initially. It is possible to conclude from this
combined analysis that the elements of reasoning in both domains interfere and
interact. Given this interconnectedness, help in either domain, either in CT and

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ST, may have had a positive impact on the thinking process as a whole.
Moreover, the CT and ST perspectives offer not only simultaneous and parallel
ways of interpreting the reasoning process, but, through a combined analysis, can
explain how CT and ST can be mutually beneficial, reinforcing related reasoning
approaches. Tasks that stimulate the use of both CT and ST are consequently of
practical importance.
However, CT and ST are not always interdependent in an obvious way, as
the following example suggests (Figure 2 and the corresponding interview
section).
Figure 2. Task “Laptops”
Interviewer: So, what do you think, with which of the following statements do you
agree?
Nena: I would agree with the third statement.
Interviewer: Can you explain, please?
Nena: Yes...400 computers is not a large sample when talking about computers
so I would go with the third statement and just listening to the comments
of her friends and the consumer’s magazine, it is possible that both the
computers could be just so.
Interviewer: Are you sure?
Nena: Sure. I like this question!
Seen from the perspective of CT, Nena questions not only the experiences of the
friends, but also the results from the study with the 400 laptops. On this basis,
she expressed agreement only with the third statement, which highlights
questioning evidence as a sub-aspect of CT. However, considering Nena’s
answers under the lens of ST, she appears not to acknowledge the statistical
power of the sample of 400 laptops. Nena remarks that it is not possible to make
a prediction on the base of the data, and she appears to compare the number of
Mrs. Blum would like to buy a reliable Laptop, either a C-Pad or an S-Top.
In a computer magazine, 400 laptops of each brand have been tested. In this
comparison the C-Pad has turned out to be more reliable.
In the evening she talks to three friends. Two have S-Tops and never had
problems. The third had a C-Pad, but had so many hardware problems with it,
that he has sold it again immediately.
With which of the following statements do you agree?
฀ Mrs. Blum should buy an S-Top, because the friend with the C-Pad had
made bad experiences, whereas the friends are happy with the S-Tops.
฀ Mrs. Blum should buy a C-Pad, because the test in the computer magazine
is based on a high number of computers, not only on one or two.
฀ No matter how she decides, it can happen that she gets a Laptop that
causes problems frequently.

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the 400 laptops to the number of all laptops. She does not reflect in depth on the
statistical power of the magazine study.
From a joint perspective, Nena’s dominant critical attitude may have blocked
her use of elements of ST, for example reasoning related to the sample size and
representativeness. This example gives insight into how CT and ST may
interfere. In this case, CT practice acted to the detriment of ST. Conversely, ST
can be dominant over CT, as the following example (associated to the problem in
Figure 3) suggests.
Figure 3. Task “Crimes”
Interviewer: …and here...Do you agree with the interpretation of the police president?
Why or why not?
Nena: I do not see a big difference with the number of crimes for any given year
that would warrant extra police force to be hired...
Interviewer: Can you justify it, please?
Nena: It looks like the average is approximately 520, which is close to all the
numbers so I do not think anything different is really happening from any
given year.
According to Ennis’ taxonomy (1989), one crucial element of CT is raising
questions, having doubts, and exploring key definitions, like “Crimes”. In this

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task, the nature of the crimes is a key consideration. For example, if we knew
that all the crimes were murders, we might decide differently than if the crimes
related to paying taxes or fraud. No question about such a definition was raised
by Nena and the focus was purely numerical. In this task, Nena employed a
restricted set of CT skills.
From the point of view of ST, Nena used an appropriate model and showed
an awareness of statistical variation. By these means, she arrived at the
conclusion, that, given the variation of the data of the past years, the rise of the
crime number is not significant. Consequently, seen from the ST perspective,
Nena showed an appropriate understanding of the statistical situation.
Looking at this part of the interview in the joint CT and ST interpretation
mode, the analysis suggests that Nena successfully questioned the statement of
the police president by using the data given in the problem and a statistical
argument. She appeared to remain in the statistical domain, giving more details
related to the model she had chosen (distances to the average value). This focus
on ST may have hindered her use of any of the CT skills, such as analysing and
questioning the definition of “crime” as the key notion here, questioning the
evidence (i.e. the way the data had been collected), etc.
Questioning data plays a role also in the following example related to the
problem in Figure 4.
Figure 4. Task “Tablets” (Kuntze et al., 2008)
A company produces two sorts of headache tablets. Both sorts have been tested in
a laboratory with respectively 100 persons suffering from headache. The diagram
below shows, how long it took until the headache was over. Each point represents
one test person.
Dr. Green:
Find counter-arguments!
Dr. Jenkins:
Find counter-arguments!
No, because ________________ No, because ________________
Tablet 1 is the better one! Tablet 2 is the better one!
Tablet 1
Tablet 2
Time in minutes

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Nena: And here…I think tablet one takes too long to get rid of the headache. Tablet
two seems to get rid of the headache a lot quicker for the majority of the
people. You don’t really know the age or weight of the people. Many factors
play into the reason why a headache might occur so the statistics are poor.
Based on the chart, I would have to pick tablet number two in hope of a
speedy recovery.
From the perspective of CT, Nena not only evaluates the given statements, but
she also shows CT elements when going beyond the data given: She gives
examples of relevant influencing factors, and questions the data provided in the
diagram (“the statistics are poor”).
From the point of view of ST, the analysis of Nena’s short answer yields that
Nena chose an appropriate model and was aware of the key elements of the
problem, even if she did not explicitly discuss the minority of cases with very
slow recovery for tablet 2. These considerations led to her personal conclusion to
pick tablet number two, as she obviously sees the chance of a “speedy recovery”
as more important than the risk of a very slow recovery.
Looking at both CT and ST, the example appears to highlight how elements
of CT can contribute to ST, for example when evaluating data, its presentation
and analysis, planning data collection, etc. In the example, Nena suggests an
analysis that takes into account the age or weight of the persons in the study.
Conversely, aspects of ST like dealing with statistical variation and uncertainty
can contribute to CT, especially when it comes to decisions in non-determinist
situations, where full data is unavailable. These examples are intended to
illustrate how both ST and CT skills can be evoked by the same task. We suggest
that this models authentic and useful thinking practice more effectively than a
more closed task that stimulated only CT and the application of taught
procedures. Our question now, is “what are the messages for task design?”
CONCLUSIONS: THE FEASIBILITY, UTILITY AND DESIGN OF
HYBRID TASKS
The tasks used in this exploratory study have certain distinctive characteristics
and we would argue that each of these characteristics constitutes a key principle
of task design:
 each uses a “real-world” situation as its “figurative context” (see Clarke &
Helme, 1998);
 each provides succinct statistical information relevant to that context
(Kuntze et al., 2008);
 the problem is stated very simply;
 some form of evaluation is integral to the problem (Ennis & Millham,
2005); and

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 the task affords many reasoning approaches.
The exploratory study demonstrates that connections clearly exist between CT
and CT at the level of individual reasoning practices. In seeking to stimulate both
forms of thinking we suggest that an individual employing CT has access to a
structured framework of analytical principles that guide and support their
reasoning. That is, the relationship between measures of central tendency and
variance, for example, structure any consideration of distribution of data that
might be invoked in drawing evidence-based conclusions or making evidence-
based judgements. On the other hand, the components of CT are not related in
such a structured fashion and an individual’s inclination to employ one strategy
(e.g., questioning evidence or questioning assumptions) can be given expression
without any obligation to also invoke other components of CT. Some CT skills
resemble the heuristics that were the focus of the enthusiasm for problem solving
in the 1980s and 1990s (Clarke, Goos, & Morony, 2007). Catalogues of such
heuristics were similarly fragmented.
Ennis and others have catalogued CT skills (Ennis, 1989) and even arranged
these categories in a form of hierarchy, but the connection between specific CT
skills is under-theorised in comparison with CT. Nonetheless, the forms of CT
identified in such classificatory schemes are clearly of significance, both as
aspects of reasoning and as potential curriculum content. If it were possible to
develop a structure for CT in which the component elements were not only
identified, but also their relationship established, then to invoke one aspect of CT
would serve to catalyse the use of other related aspects, because the connections
between elements would be well known and understood. The question of how
best to conceptualise these skills, how to integrate or connect them with other
curricular goals, and how best to promote them and nurture their development in
the classroom has been a major challenge. An earlier study (Aizikovitsh-Udi,
2012), using similar tasks, has documented efforts to produce CT through a
program of instructional immersion in the related topic of probability. In this
paper, we argue that particular tasks can stimulate the use, promotion and
development of both Statistical and CT. We would like to suggest that an
instructional program of hybrid tasks could be devised that provides the
opportunity to employ CT, while simultaneously introducing students to the
practices and structure of CT. The design characteristics of such hybrid tasks
have been identified.
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Academic Publishing.

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classrooms around the world (pp. 185-216). Rotterdam, The Netherlands:
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This document was originally published as Aizikovitsh-Udi, E., Clarke, D., &
Kuntze, S. (2013). Hybrid tasks: Promoting statistical thinking and critical
thinking through the same mathematical activities. In C. Margolinas (Ed.), PTask
desing in mathematics education. Proceedings of ICMI Study 22 (Vol. 1, pp.
453-461). Oxford, United Kingdom: International Commission on Mathematics
Instruction.
Einav Aizikovitsh-Udi
Beit Berl College
einav.aizikovitsh@beitberl.co.il
David Clarke
University of Melbourne
d.clarke@unimelb.edu.au
Sebastian Kuntze
Ludwigsburg University of Education
kuntze@ph-ludwigsburg.de
Received: December 2013. Accepted: January 2014.
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