LESSONS FROM INFERENTIALISM FOR
STATISTICS EDUCATION
ARTHUR BAKKER
Freudenthal Institute, Utrecht University
a.bakker@fi.uu.nl
JAN DERRY
Institute of Education, University of London
j.derry@ioe.ac.uk
ABSTRACT
This theoretical paper relates recent interest in informal statistical inference (ISI) to
the semantic theory termed inferentialism, a significant development in contemporary
philosophy, which places inference at the heart of human knowing. This theory assists
epistemological reflection on challenges in statistics education encountered when
designing for the teaching or learning of ISI. We suggest that inferentialism can serve as
a valuable theoretical resource for reform efforts that advocate ISI. To illustrate what it
means to privilege an inferentialist approach to teaching statistics, we give examples
from two sixth-grade classes (age 11) learning to draw informal statistical inferences
while developing key concepts such as center, variation, distribution, and sample without
losing sight of problem contexts.
Keywords: Inferentialism; Representationalism; Statistical inference; Statistical key
concepts; Web of reason
1
A major reason for people to use statistics is to generalize from a sample to a population, a
process, or a future trend. People who do not (yet) have formal inferential techniques at their
disposal might still need to draw statistical inferences in their daily life or at work (Bakker et
al., 2008). Because of the relevance of being able to draw data-based conclusions while
acknowledging uncertainty and variability, informal statistical inference (ISI) has become a
central topic in statistics education research (Ben-Zvi, 2006; Pfannkuch, 2006; Rubin,
Hammerman, & Konold, 2006; Watson & Moritz, 1999). It can be characterized as a
probabilistic generalization based on the evidence of data (Makar & Rubin, 2009).
Even though statistics educators have made progress on the questions of what ISI is and
how it might be promoted in education, this area of research could in our view benefit from a
stronger theoretical foundation rooted in what it means to know something about a context of
inquiry, and, on the basis of such foundation, from further insight into the challenges faced
when fostering ISI at the school level (summarized in Section 1). Using a significant
development in philosophy, that of a semantic theory called inferentialism (Brandom, 2000),
the goal of this paper is to articulate three lessons related to three challenges we faced in our
own research in which informal statistical inferences were promoted (Bakker, 2004a), and to
underpin theoretically current research on ISI (Section 2).
Inferentialism attends to the distinctive nature of human awareness and puts inference at
the heart of human knowing by providing an account of concept use that starts with reasoning
rather than with representing. Due to this emphasis we think inferentialism has the potential
to theoretically underpin efforts to support educational research on ISI but also on statistics
and mathematics more generally. In brief, Brandom characterizes his theory as pragmatist,
holist, rationalist expressivism. Derry (2004, 2008) argues that Brandom’s theory is
2
compatible with Vygotsky’s ideas, due to the influence of Hegel. Although several of the
lessons we draw from Brandom resonate with those made by educational thinkers (e.g.,
Bruner, 1983; Sfard, 2008), his work is particularly interesting as it probes the nature of
awareness itself and has an explicit focus on reasoning (i.e. inference) underpinning concept
use. Moreover, inferentialism helps to explore the relationships between statistical inference,
concepts, and context—relationships that hitherto have only been touched upon in recent
literature on ISI. We illustrate these lessons from inferentialism with examples from teaching
experiments in Grade 6 in which students learned to draw informal statistical inferences
(Section 3). Finally we reflect on the main issues brought up in the paper (Section 4).
1.
THREE CHALLENGES IN STATISTICS EDUCATION
Based on our experience with stimulating ISI (Bakker, 2004a) we think that any teacher,
educator, or designer who tries to support students’ informal inferential reasoning
(Pfannkuch, 2006) leading to informal statistical inferences is likely to face at least three
related challenges that are persistent in education in general and manifest themselves in
statistics education in specific ways.
1. The challenge to avoid inert knowledge—knowledge students have learned to
reproduce but cannot use effectively (Bereiter & Scardamalia, 1985; Whitehead,
1929).
2. The challenge to avoid atomistic approaches found in many textbooks and to foster
coherence from a student perspective.
3. The challenge of sequencing topics in alternative approaches which aim for
coherence from a student perspective.
3
Challenge 1. Even if students have learned the main statistical concepts and graphical
displays, they often fail to use them to solve statistical problems. The best documented
problem in this area is that students who have learned the arithmetic mean tend not to use it
for comparing two groups (Konold & Higgins, 2003; McGatha, Cobb, & McClain, 2002;
Pollatsek, Lima, & Well, 1981). In other words, we expect students to use the mean (or any
other concept) within a process of reasoning, but they often use it purely descriptively or just
calculate it out of habit. Mokros and Russell (1995, p. 37) go so far to argue that “Premature
introduction of the algorithm … may cause a short circuit in the reasoning of some children.”
The challenge is therefore to stimulate inferential reasoning in such a way that students
naturally use the mean and other statistical concepts, for example, when comparing groups.
Challenge 2. Many statistics textbooks address lists of topics that together form the
toolkit with which students can solve statistical problems. At the middle school level, for
example, students are introduced to the mean, median, mode, bar graph, histogram, box plot
and so on; at higher levels, students might learn series of more advanced techniques such as ttest and F-test.
Reform attempts such as those advocated by Cobb (1999, 2002) aim to provide more
coherence in the curriculum and argue that statistical concepts and graphical displays should
be taught in relation to each other. One way to do so is by supporting students’ understanding
of the distribution of a data set at the middle school level (see also Bakker, 2004b; Cobb,
McClain, & Gravemeijer, 2003; Petrosino, Lehrer, & Schauble, 2003; Russell & Corwin,
1989) and even primary school level (Ben-Zvi & Sharett-Amir, 2005) with such graphical
representations that students can draw data-based conclusions. In line with the recent
literature on ISI we take ISI to be central in statistics education and treat key concepts such as
4
distribution, data, center, variation, and sampling as reasoning tools for drawing data-based
conclusions about some wider universe.
More generally, well-known approaches to improve coherence from a student perspective
and foster a more holistic approach have been problem-based learning (Hmelo-Silver, 2004),
project-based learning (Barron, 1998), the problem-posing approach (Klaassen, 1995;
Kortland, 2001), and inquiry-based learning (Edelson, Gordon, & Pea, 1999). In these
approaches, a problem, project, or inquiry is the driver of learning the knowledge relevant to
the case in a coherent way. Attempts to teach ISI are likely to benefit from such approaches.
Challenge 3. However, the aforementioned approaches, which do not necessarily follow
the disciplinary hierarchy of introducing concepts and techniques, are faced with a third
challenge: how to sequence topics and give students access to concepts that are often
considered too difficult for them to engage with. In order to characterize the presuppositions
of common practice in schools, we contrast the research on distribution mentioned above
with the position of Batanero, Tauber, and Sánchez (2004) regarding an introductory
statistics course at university level on normal distribution:
It is important that students understand basic concepts such as probability, density curve, spread
and skewness, and histograms before they start the study of normal distribution; its understanding
is based on these ideas. (p. 275)
Taken literally, as a premise for the teaching of statistics or mathematics, the statement
implies that most of our school students will never learn about the normal distribution even
though many of them will need some minimal and not necessarily formal understanding of it
in the future in order to draw conclusions about, for instance, a production process (Bakker et
al., 2008; Pyzdek, 1991). The question is therefore how students can use the power of
5
statistical concepts or their informal precursors in making inferences prior to appreciating
formal definitions.
The hierarchy of concepts seemingly claimed in the quote raises questions. For example:
if the scientific definition of concept D builds upon concepts A, B, and C, should A, B, and C
then be taught before D? The empirical studies cited above with young students seem to vote
against such a view, because awareness of distribution is intricately connected to awareness
of center, spread, skewness, and shape as represented in dot plots, histograms, or box plots.
Moreover, as we just mentioned, some understanding of key concepts such as the normal
normal distribution will come too late for students who leave school at the age of 16.
We argue that Brandom’s (1994, 2000) position that inference is a necessary and inseparable
part of using concepts, is particularly helpful in thinking through problems relating to the
design of teaching approaches to ISI. Working out this position in the next section we draw
three lessons from inferentialism. As a word of warning we should emphasize that the term
“inferential” when used philosophically has a broader meaning than when used in statistics,
referring to the commitments implicit in any concept use. For instance, committing myself to
a particular animal being a dog precludes me from committing myself to its being a mouse. In
statistics, the term inference generally refers to an explicit and conscious process of reasoning
from a sample to some wider universe. To mark this difference, we use the adjective
“statistical” in front of “inference” whenever we need to distinguish it from the more general
philosophical usage. Likewise the term “representation” in the philosophical sense is broader
than in statistics. Where statistics educators might mainly think of graphical representations,
the term in philosophy is used more widely, for example for signs, labels for concepts,
6
formulas, models, written and spoken statements etc.—anything that stands in place of
something else (i.e. anything that is re-presented).
2. THREE LESSONS FROM INFERENTIALISM
The research cited on ISI advocates that students learn to make probabilistic
generalizations from data (Makar & Rubin, 2009). This requires, among many other things
(Makar, Bakker, & Ben-Zvi, this issue), the development of statistical concepts. But should
they learn the key statistical concepts before they can reason with them, or should they first
be invited to draw conclusions? Or is there some intricate interplay between learning to
reason and developing concepts? In this section we highlight particular ideas arising from
inferentialism that shed light on these questions.
2.1. UNDERSTANDING CONCEPTS PRIMARILY IN INFERENTIAL TERMS
The first lesson we draw from inferentialism is that concepts should be primarily understood
in terms of their role in reasoning and inferences within a social practice of giving and asking
for reasons, and not primarily in representational terms. We clarify this point in Brandom’s
philosophy in two steps. First, we summarize what can be characterized as
representationalism and what Brandom means by inferentialism. Second, we recapitulate his
line of argument, explaining the meaning of concepts and representations in terms of the
inferences that constitute them.
2.1.1 Representationalism and Inferentialism
7
Representationalism refers to the position that representations are the basic theoretical
construct of knowledge. In common with several philosophers (e.g., Dewey, Heidegger,
Rorty, Wittgenstein) and educators (e.g., Cobb, Yackel, & Wood, 1992), Brandom (2000)
takes issue with this approach noting the dominance of the representational paradigm since
Descartes:
Awareness was understood in representational terms (…). Typically, specifically conceptual
representations were taken to be just one kind of representation of which and by means of which
we can be aware. (p. 7)
Representationalism is based on the assumption that the use of concepts was explained by
what they refer to (i.e where conceptual content is primarily understood atomistically rather
than relationally). Knowing what individual concepts mean is then the basis for being able to
make sentences and claims, which in turn can be connected to make inferences. Assuming
that a definition of a concept fully conveys its meaning is a possible consequence of such a
view.
Brandom reverses the representationalist order of explanation, which leads to an account
that he refers to as inferentialism. Taking judgments as the primary units of knowledge rather
than representations, he reminds us that:
One of [Kant’s] cardinal innovations is the claim that the fundamental unit of awareness or
cognition, the minimum graspable, is the judgment. Judgments are fundamental, since they are the
minimal unit one can take responsibility for on the cognitive side, just as actions are the
corresponding unit of responsibility on the practical side. (…) Applying a concept is to be
understood in terms of making a claim or expressing a belief. The concept concept is not
intelligible apart from the possibility of such application in judging. (Brandom, 2000, pp. 159-160)
This entails giving priority to inference in accounts of what it is to grasp a concept:
8
To grasp or understand (…) a concept is to have practical mastery over the inferences it is
involved in—to know, in the practical sense of being able to distinguish, what follows from the
applicability of a concept, and what it follows from. (Brandom, 2000, p. 48)
This clarifies his definition of concepts as “broadly inferential norms that implicitly govern
practices of giving and asking for reasons” (Brandom, 2009, p. 120). Any inference leading
to a claim is made within such a normative context.
Claims both serve as and stand in need of reasons or justifications. They have the contents they
have in part in virtue of the role they play in a network of inferences. (Brandom, 2000, p. 162)
The implications for education become clearer when we consider Brandom’s vivid example
of what it is to be a concept user by contrasting a thermostat’s reliable responsive disposition
to turn on a furnace when it is cold with that of a human (knower) who can also reliably
respond to cold by turning on the furnace. Brandom asks,
What is the knower able to do that …the thermostat cannot? After all they may respond
differentially to just the same range of stimuli…The knower has the practical know-how to situate
that response in a network of inferential relations—to tell what follows from something being …
cold, what would be evidence for it, what would be incompatible with it, and so on (2000, p. 162,
emphases in the original).
The knower is capable of making a judgment, whereas the thermostat is not, and this
distinguishes the response of the knower from that of the thermostat. The thermostat’s
response is simply a moment in a series of causal stimuli whereas human responsiveness
involves reasons, not merely causes. This is relevant to teaching given that our aim in
education is not to produce automatons such as thermostats with a reliable disposition to
respond to a stimulus, but intelligent students who can make proper judgments in nonstandard contexts.
9
In the case of a concept user who has the know-how involved in using a concept, we
could speak of a state of knowledgeability, but for a fledgling knower (a learner of a concept)
what occurs in concept use is a process of getting to grips with the content of a concept and
the conditions of its application (this is precisely what an infant does in mastering the use of
words and thus learning their meaning—see Bruner, 1983). In order to do this the learner
needs experience in the “space of reasons”—a philosophical term Brandom borrowed from
Sellars (1956/1997)—in which the concept is used. Brandom (1994, 2000) and McDowell
(1996) use the term space of reasons for the whole space of reasons in which humans live.
Brandom (1994, p. 5) uses the more modest term “web of reasons” for particular situations.
As a philosophical concept the “web of reasons” can help us think about the complex of
interconnected reasons, premises and implications, causes and effects, motives for action,
utility of tools for particular purposes, that have rational impact in a particular context. Even
though people may not be consciously aware of the particular web of reasons in play in the
context in which they are active, for example when drawing an inference about a production
process in a car factory, those reasons still impact upon their work (Bakker et al., 2008).
To clarify the concept of web of reasons, we give an example involving statistical
knowledge. Assume a fish farmer wants to know if genetically engineered (GE) fish grow
bigger than normal fish. The reason (1) for being interested in this is that GE fish might turn
out to be more profitable. To see if the GE fish are longer than the normal ones (reason 2), he
decides to throw fingerlings of both types into a pond and catches a sample after a few
months. To see if the means are fair group descriptors (reason 3), he looks at the data
distributions. To get a sense of the effect size (reason 4), he also calculates the standard
deviations of both groups. This example thus involves a web of many reasons, of which we
10
made only a few explicit. A knowledgeable person is responsive to a rich web of reasons in
such contexts. Some reasons might be statistical (the fish farmer might even want to do a ttest because this is more reliable) and others contextual (s/he may need more money or the
presence of GE fish might have affected the growth of the normal fish), but the main point is
that these reasons are relevant due to their inferential connections, and their bringing to light,
in turn, depends upon the judgments of the fish farmer i.e. his own responsiveness to reasons.
2.1.2 Explaining the meaning of representation in terms of inference
Given the priority of inference over representation in the use of concepts, Brandom
(2000, p. 39) takes “up the challenge of explaining the referential or representational
dimension of concept use and conceptual content in terms of the inferential articulation.”
This does not diminish the importance of representation, because evidently “there is an
important representational dimension to concept use.” (p. 28)
There is not the space here to rehearse Brandom’s line of argument in detail; in short, he
argues that semantics is rooted in pragmatics. Meaning should be understood as constituted in
activity or social practice:
the representational dimension of discourse reflects the fact that conceptual content is not only
inferentially articulated but also socially articulated. The game of giving and asking for reasons is
an essentially social practice. (p. 163)
Brandom’s line of argument leads to the first lesson we want to draw with respect to the
learning of statistical concepts and inferential reasoning: it is in the context of reasoning (the
distinctively human responsiveness to reasons, as in the thermostat example above) that
representations (words, graphs, inscriptions etc.) gain and have meaning. In particular,
statistical concepts such as mean, variation, distribution, and sample should be understood in
11
terms of their role in reasoning, i.e. in terms of the commitments entailed by their use. As we
argue here, learning statistical concepts in the practice of making inferences about and from
data sets makes visible the commitments integral to their appropriate use. In relation to
Challenge 1 of Section 1, to avoid inert knowledge, we should therefore look for ways to
emphasize the inferential function of statistical concepts and graphical representations in
order for them to gain meaning for students and become productive. More specifically we
advice to introduce statistical concepts and graphical representations in the context of making
inferences about what students take to be realistic problem situations.
2.2. PRIVILEGIZING HOLISM OVER ATOMISM
A further distinction that Brandom (2000) makes to characterize his inferentialism is that
between atomism and holism. Representationalism seems to lead to an atomistic approach as
the meanings of concepts are assumed to be understandable solely by their reference to what
they represent and then are combined to make judgments and inferences. By contrast
inferentialism is resolutely holistic: “one cannot have any concepts unless one has many
concepts. For the content of each concept is articulated by its inferential relations to other
concepts. Concepts, then, must come in packages” (Brandom, 2000, p. 15-16) and so
“[c]ognitively, grasp of just one concept is the sound of one hand clapping.” (p. 49) Our
second lesson—that one cannot inferentially reason with any concept without drawing on its
inferential relations to other concepts—is a corollary of the first. This point of privileging
holism over atomism might initially appear trivial, but it has significance for Challenge 2 for
avoiding atomistic approach to statistics education.
12
In practical terms, we consider it important for teachers to provide students with
(informal) ways to attend to other relevant aspects of data sets such as variation, distribution,
and sample size in addition to learning definitions and calculations (cf. Bakker, 2004a; Cobb,
1999; Konold et al., 2002; Makar & Confrey, 2005; Zawojewski & Shaughnessy, 2000).
2.3. PRIVILEGING AN INFERENTIALIST APPROACH
For Vygotsky (1998) the development of concepts proceeds through activities in which the
concepts function meaningfully. Hence a concept is not first learnt formally and then applied,
but develops according to the domain of activity (including reasoning) in which it functions.
The inferentialist view presented above may initially appear counterintuitive. Certainly,
some theories of language acquisition as well as philosophy of language have relied on the
idea that the simple terms must be learnt prior to those which are more complex, or prior to
the combination of such terms in judgments before one can reason with those terms.
However, Vygotsky alerts us to the idea that it is possible to operate with a concept before
fully understanding its meaning, for example for a child to begin to operate with complex
structures before fully grasping simpler ones. The reason for this is that knowledge is not
gained simply by an accretion of elements starting with the most simple. In the case of
statistics education we can think of distribution as a complex structure and mean as a simpler
one. Yet understanding the mean requires some sense of distribution because, for example,
using the mean for comparing distributions depends on their shapes: with unimodal and
roughly symmetric distributions, using the mean is commonly accepted, but with outliers or a
very skewed distribution one might prefer the median.
13
In Vygotsky’s understanding of language use, a child is inducted into correct application
of concepts, and hence meaning, via activities with others within a normative practice. The
meaning of any concept is not accessible immediately, all at once, but arises out of a
developmental process involving a fine-tuning imposed by the norms governing application
of concepts, namely the inferential relations that form the content of the concept in the first
place. Likewise any application of a concept involves and draws upon a whole series of
judgments and actions, whether or not the concept user is explicitly aware of the judgments
and actions involved.
Vygotsky stresses this point when he speaks of “a system of judgments” and argues that
the idea of “general representations” is inadequate to express what a concept is in reasoning.
In line with our first lesson about the priority of inference over representation, he argues that:
we must seek the psychological equivalent of the concept not in general representations, … not
even in concrete verbal images that replace the general representations – we must seek it in a
system of judgements in which the concept is disclosed. (Vygotsky, 1998, p. 55)
Contrary to a view that takes meaning to be based on a representational relation to the
world, Vygotsky (like Brandom) privileges an inferentialist over a representationalist view:
“To think of some object with the help of a concept means to include the given object in a
complex system of mediating connection and relations disclosed in determinations of the
concept.” (Vygotsky, 1998, p. 53)
The inferentialist view alerts us to the normative character of concept use. What counts as
valid reasoning, adequate judgment, or correct application of concepts depends on the norms
being used in a particular practice. In the case of formal statistical inference, these norms
have become so common that most of us are hardly aware of them anymore when drawing
14
statistical inferences. However, when working with learners it is clear that these norms still
have to be established (cf., Cobb, 1999; Yackel & Cobb, 1996).
Based on the representationalist and inferentialist views elaborated above, we can
distinguish two approaches to statistics education that are in line with these views. While the
representationalist view may not be found in the research literature in its extreme form, we
still consider it useful to remind practitioners of the problems with this view, as a number of
factors (time and resource pressures, curricular constraints) militate against giving the
necessary weight to pedagogy. An approach that privileges representations focuses on
concepts and graphs as representations, abstract mirrors of some reality—whether physical or
ideal. This representationalist approach lends itself to a “topic-by-topic approach” in which,
for example at the middle school level, mean, median, mode, standard deviations, histograms,
and box plots are taught atomistically in separate sections from simple to more complex. It is
assumed that once students have learned the definitions and procedures, they will be able to
solve a statistical problem by applying what they have learned. In this representationalist
approach, it is generally not recognized that definitions stand in place of (‘re-present’)
statistical concepts and graphs that refer to situations arising within a social practice of
people who already know about statistics and therefore are already aware of the relevant
inferential connections between the concepts and contextual factors that constitute the
representations in the first place (cf. Roth & Bowen’s, 2003, work on scientific graphs).
In an inferentialist approach, it is assumed that grasping a concept or understanding a
graph is mastering the use of the word or graph in a process of reasoning. Privileging an
inferentialist approach over a representationalist one is not to say that one does not need to be
precise or work towards conventional definitions. On the contrary, because the inferential
15
game involves articulating reasons (Brandom, 2000) and making things explicit (Brandom,
1994) within a social practice, this necessarily requires being specific and accurate in what
the students and teacher refer to, i.e. making explicit what is entailed by the use of a concept
and what it follows from. There is a definite need to come to clear delimitations of concepts
to allow meaningful communication.
For the purpose of providing access to concepts or types of reasoning normally
considered too difficult for learners (such as the normal distribution and statistical inference),
the order of teaching should not mainly be governed by the disciplinary structure of statistics
as typically represented in textbooks, but by the inferential function of the concepts at stake,
more specifically those reasoning steps that lie in students’ zone of proximal development
(Vygotsky, 1978).
2.4. SUMMARY
In the previous sections we have formulated the following three lessons from
inferentialism. First, concepts should be primarily understood in terms of their role in
reasoning and inferences within a social practice of giving and asking for reasons, and not
primarily in representational terms. Brandom summarizes this point as privileging inference
over representation. Second, to understand concept use we should privilege holism over
atomism. Third, based on these philosophical lessons we put forward an educational
approach that we call an inferentialist approach in which these two lessons are translated to
the discipline of education. This third lesson is summarized as privileging inferentialist
approach to education over a representationalist one.
16
In the next section we give empirical illustrations of the three lessons drawn above, in
particular of an inferentialist approach on learning to draw statistical inferences while
developing statistical key concepts such as distribution, center, variation, and sample. These
illustrations will hopefully show that the theory articulated in this section is not just an
academic debate but has real consequences for education.
3. EMPIRICAL ILLUSTRATIONS
3.1. ORIGIN OF THE EMPIRICAL EXAMPLES
The empirical illustrations stem from teaching experiments in Grade 6 (most students were
11 years old). These were carried out in the Midwest of the USA as part of the TinkerPlots
design project (Konold, 1998) and as part of the revision of the Dealing with Data unit of the
Mathematics in Context curriculum for middle school level (De Jong, Wijers, Bakker,
Middleton, Simon, & Burrill, 2006).
The purpose of the teaching experiments was to revise instructional materials in order to
take
advantage
of
the
TinkerPlots
capabilities
(Konold
&
Miller,
2005—
www.keypress.com/x5715.xml), but also to provide feedback to the TinkerPlots designers
who were then still developing it. With this software we hoped to engage students in
statistical reasoning and the topic-by-topic approach that was apparent in the students’
textbook, but we also assisted the teacher in supporting her students in their learning with the
software. Many tasks involved the comparing of two data sets in order to stimulate the use of
averages as group descriptors. In line with Bakker’s (2004b) positive experiences with
growing samples, tasks around the fish story (Sections 2.1 and 3.3) were used to stimulate
informal inferential reasoning with multiple statistical concepts in connection to each other.
17
The teaching experiments were carried out in two classes with 22 students each and
lasted for respectively 18 and 20 periods of 42 minutes each. Class periods were videotaped,
interviews with students and their teacher were audio recorded, extensive daily reports were
written about observations made that day and relevant parts of the recordings were
transcribed. At the end of the school year the teacher spontaneously wrote a four-page
summary of her views on the two teaching experiments. We quote from this summary to
contrast her initial rather representationalist approach to statistics education with the final,
more inferentialist approach.
3.2. FROM A REPRESENTATIONALIST TO AN INFERENTIALIST APPROACH
The school and teacher’s approach before the teaching experiments could be characterized as
representationalist. Students had learned to find mean, median, and mode in grade 4 or 5, but
in the first period of the teaching experiments in Grade 6 it became clear that they had very
little idea of what these measures were or how to use them in answering the question of
whether boys’ feet were longer than girls’ feet. In representational terms, these students did
not really know what mean and median were, and, in inferential terms, they had not grasped
the reasoning informing each concept either: they did not know how to come to a conclusion
(inference) using their knowledge of the median values of boys and girls.
The teacher, in retrospect, wrote the following about her initial approach (this was
without our prompting or discussing):
In my data unit my whole focus seemed to be on the importance of creating a good graph, the
ability to read graphs that were given to students and to be able to identify the range, mean,
median, and mode. … I was always frustrated as a teacher that students could one day state the
mean, median, or mode but the next day mixed up terms and so the majority of discussion seemed
18
to be towards drilling how you find mean, median, and mode, and little about what is the reason
we are looking for these values and what they could tell you. This seemed important to me as a
science teacher. I found students writing the mean, median, and mode in their conclusion of their
experiments and yet rarely seemed to include information about why they would want to know this
or reflect on why they may see the results they saw.
Although she felt that statistical information should be used to come to a conclusion, several
of her statements suggest that she felt tied up in what we call a representationalist approach,
for example “creating a good graph” (a very common focus according to Friel et al., 2001)
and “identify the range, mean, median, and mode.” What seems implicit is the teacher’s
assumption that once students have learned to produce these graphical representations and
procedures, they can reason with them.
The teacher and first author felt they needed to engage these students in statistical
reasoning about relevant features of data sets to be able to draw conclusions about and
beyond the data. In other words, we strove for what we now call a more inferentialist
approach. We intend to illustrate what it takes for students to understand some key concepts
inferentially (for example to compare two groups) rather than just representationally. After
describing activities that involved students in reasoning about statistically relevant aspects of
data sets we illustrate how the three philosophical lessons might play out in practice.
3.3. EMPIRICAL ILLUSTRATIONS OF THE THREE INFERENTIALIST LESSONS
As a background to the fourteenth period from which we take our illustrations, we should
share a few observations from the periods before. One is that students used what Vygotsky
calls “spontaneous” (as opposed to scientific) concepts or precursors to concepts that support
students’ inferences when they do not yet have a fuller grasp of the scientific concepts. We
19
discuss their notion of a “clump” as an example. This notion emerged in the twelfth period
from using terms such as “clumped,” “clustered,” and “stacked” (cf. Bakker, 2007) as a way
to compare two groups, and the researcher and teacher stimulated students to indicate where
exactly they saw these clumps in the data sets. The motivation to do this was that Konold et
al. (2002) had noticed that reasoning about “modal clumps” could well be a precursor to
using measures of center and spread when comparing data sets (see also Bakker, 2004a;
Makar & Confrey, 2005) and based on the earlier periods the sixth graders were judged not to
be ready to use the conventional measures of comparing data sets (say with mean, median, or
interquartile ranges).
The problem situation we used in the fourteenth period to support students’ statistical
reasoning was the fish farmer story, originally designed by Konold and Pollatsek (2002). Just
like the example used in Section 2.2, a fish farmer claims that his genetically engineered
(GE) fish grow bigger than his normal fish. One year after releasing a bunch of normal
fingerlings and a smaller number of GE fingerlings into a pond, students are allowed to catch
some fish to check his claim. Each student simulated “catching” about four fish from the
pond by drawing cards from a box. The type and length (in cm) of each fish was shown on
the data card and each student had an activity sheet with axes on which they plotted their own
data as well as the data that other students read aloud.
normal
20
genetic
Figure 1. Samples of normal and genetically engineered (GE) fish lengths in cm on the
whiteboard
The reason we used this “growing a sample” activity was that previous research (Bakker,
2004b) indicated that it can engage students in drawing conclusions rooted in awareness of
multiple relevant statistical aspects of data sets such as center, variation, distribution, and
sample size. We did not teach these aspects atomistically one by one, but took a holistic
inferential approach (Lesson 2). The primary question under discussion was whether GE fish
tended to grow longer than normal fish. Part of the discussion was first to understand this
question—establishing that we did not mean by this that all GE fish were longer than all
normal fish. So even in establishing a ‘reason’ for students to investigate the data, there is a
need to develop norms or understandings about what we mean by the question, and what we
accept as evidence for the claim. Concerning the latter, there was an implicit assumption that
data gathered from the pond would constitute evidence of this claim, an assumption that all
students appeared to accept. Note that in drawing the fish, we had students draw without
looking, thereby instituting a norm we did not explain—that it is important for our samples
not to be biased. This is an example of how we were introducing students into a normative
practice and thereby not having them explore all issues at once. Some ideas and issues are
problematized while others are backgrounded, for the moment.
21
To involve all students, we asked them to take turns going up and plotting their own data
on the whiteboard, while others plotted all data points on their own sheets. After a couple of
turns, students collectively had a closer look at the intermediate result. With regard to Figure
1, we first asked for observations about the data sets. The reason for doing so was that
observations would involve students in making judgments which are part of the web of
reasons that we wanted them to operate in.
Many students used the term “clump,” probably referring to the majority of values in the
middle – a notion that had emerged the day before when comparing two data sets. We aim to
illustrate here how this spontaneous concept gained meaning in a reasoning process (Lesson
1) and how it might function as a precursor to the more conventional mean as a group
descriptor when comparing groups. We consider the clump notion a precursor to the mean
because it is inferentially used in a similar way to how the mean or quartiles are used for
drawing conclusions about two groups (Lesson 3).
3.3.1 Illustration 1: Privileging inference over representation
As part of the class discussions, we often had to ask students where exactly they saw the
clumps in the two distributions of Figure 1, thus pushing the representational dimension of
the reasoning practice. Students would then answer, for example: “The clump [of the normal
fish] is in between 18 and 23 [cm].” For even easier reference, we sometimes asked students
in addition to indicate the clumps with dividers or reference lines in TinkerPlots. One reason
why this is important is that the range of the clump has the potential of drawing students’
attention to a particular type of variation—not just as a figurative feature of a representation
per se—thus enabling inferential reasoning with awareness of variation in data.
22
One girl used the clump notion as a measure of both center and variation to compare the
two types of fish while taking the sample size into account. She thus described aspects of the
data distributions that were relevant to drawing a conclusion – part of the web of reasons she
was operating in.
Linda:
Right here [pointing to the normal fish data in Figure 1] there is a bigger clump than
this one [GE], but the numbers are less. This one [normal], the clump is more spread
out but higher.
In this statement, it is quite unclear what she means by such terms as “less” and “higher” as
there are two potential quantities she could be referring to: the number of cases in a clump,
and the location of the clump along the axis. So the researcher (Res.) asked her to repeat her
claims, and even though she was still not quite explicit, we think we can now understand her
claims (as indicated by our interpretations in the brackets).
Res.:
You said a lot of things. So one was?
Linda:
This [normal] clump is bigger [has more values] but has a smaller number [is located
lower on the axis]. And this one [GE] is smaller [fewer values], but it has a higher
number [located higher on the axis].
The example illustrates how students work towards developing concepts with which they
can make explicit their inferential reasoning rooted in growing awareness of different types
of variation: between the sample sizes (the GE fish form a smaller sample, which also causes
the lower height of the clump) and between the location of the two distributions (GE fish tend
23
to be longer). To support their claim that the GE fish were bigger than the normal fish, the
driving question, students needed to clarify what they meant. And in support of this process,
we had to keep asking what exactly they referred to—the representational dimension of the
reasoning process: where do you see the clump, what do you mean by “higher” or “more”?
Students even started to query each other:
Karen:
If you say “clump,” do you mean where most of the fish are? Or do [interrupted].
Kerry:
Yeah, like where most of the fish are normal; their length is in between the range
28–35, where most of them are.
Though giving such priority to reasoning over representing takes time, we argue it is
necessary to pay attention to those observations of aspects of data distributions (cf.
Pfannkuch, 2006). Even if the end result is that students compare the means of the two data
sets when making a statistical inference, the web of reasons they will operate in will be more
extensive than when they were trained to compare means. This brings us to the second
inferentialist lesson.
3.3.2 Illustration 2: Privileging holism over atomism
Our aim with this learning activity was that students would look at the whole distribution
before choosing a measure of center or spread for comparing two groups. And indeed,
students such as Linda (above) were generally aware of many features of the distributions
(e.g., variation, center, sample size) in relation to each other (Lesson 2), particularly when
hypothesizing on features of larger samples (cf. Bakker, 2004b).
24
After discussing the shape of the two distributions and students’ predictions, we returned
to the original question of whether the GE fish grew longer than the normal fish.
Anissa:
I would say that there is more of a variety in the normal fish, but I would actually say
that the genetic fish do grow taller, because more of them are closer to the right
side.… So I think he [the fish farmer] is right.
Tom:
If you look at the range, it kind of goes from like 18, … it goes from 18 to 35, and
for the normal it goes from about 8 to 37, but there is more of the, there is more
genetic fish towards the right.
Our experience was that students often mentioned aspects of variation before they talked
about differences in size. This finding is consistent with the findings of Konold et al. (2002).
We conjecture that if we as statistics educators do not push students to use mean, median, or
mode all the time as if these are the main things to learn in Grades 5-7, but use instructional
activities that ask for students’ observations about variation as well, students will display
good awareness of different types of variation.
3.3.3 Illustration 3: privileging an inferentialist approach
The last illustration shows what it means to privilege the inferential function of concepts,
in particular how some students used the mean inferentially as part of a system of judgments.
The teacher and first author never told students to use mean or median values to compare two
groups nor encouraged them to use such measures of center, though the measures were
available in the software at the click of a button.
25
The question in this case was whether the fish farmer was right in claiming that GE fish
were about twice as big as the normal fish, instead of just bigger. Building on their prior
reasoning with clumps, this spontaneous concept of clump became a tool for reasoning about
aggregate features of data sets. Note that asking about the size of a difference in general
requires a different set of justifications than simply deciding whether the groups differ at all.
We purposefully used this question to push the students towards more precise and
quantitative descriptions of center.
Using TinkerPlots, Tom performed a series of actions that had become habitual over
time. He first separated the dots representing the different fish types vertically, ordered the
lengths horizontally, stacked the dots, and used the mean button and reference lines to
compare the types of fish (Figure 2). He explained further:
I clicked the mean value and the reference line, because it shows kind of where the clump is. And
that helps me because it is easier for me to see where most of them are. And this one [the normal
fish] there is a lot more, there is 292 [using the count option]. And this one, there is 67, so there is
about a fourth, a little under a fourth. This one [GE] is a lot more gradual, it is spread out, but they
grow a lot bigger and this one [normal] is very steep [points with the mouse along the slope] and
then is really steep, going down. And you can see that it is not really twice. These ones aren’t
twice the size of these ones. It is more like one and a half times.
26
Figure 2. The graphs Tom used to explain his statistical inference that the GE fish were
about 1.5 times the normal size. The blue triangles give the position of the mean values, and
the movable vertical red lines are called reference lines.
Note that Tom naturally included reasons in his explanation. This quote thus shows part
of the web of reasons in which Tom made relevant observations, judgments, and actions
before he arrived at his conclusion about the contextual question. For example, it is
promising that Tom used the means to indicate where the clumps were while taking into
account the different types of variation—variation between and within data sets as well as
variation in sample size and perhaps even variation around an ideal shape.
To illustrate that the teacher felt she had moved to a more inferentialist approach, we
quote from her end-of-year reflections on the teaching experiments: “I heard discussions that
were shifting to what the data might be telling us and less on a reciting of this is the mean,
median, and mode (often identifying the wrong value).” And:
27
I found discussions led by me were becoming more centered around the shapes of graphs, the
amount of variation and the clumps or clusters. These discussions seemed richer and students
appeared to be closer to using data to answer questions about what the data is saying.
In other words: students drew inferences about a contextual problem while attending to
several statistical aspects such as distribution, center, and variation. We assume that what
contributed to students’ abilities to draw sensible inferences in our illustrations was an
inferentialist approach (Lesson 3) in which we privileged inference over representation
(Lesson 1) and holism over atomism (Lesson 2).
4. DISCUSSION
In this paper we suggested that anyone who designs for the teaching and learning of drawing
informal statistical inferences is likely to encounter the challenges mentioned in Section 1.
We think that inferentialism provides a useful theoretical language to articulate the
challenges in terms of representation and inference. In this discussion we reflect on the three
challenges, which are in fact not bound to statistics education, and the three inferentialist
lessons that we illustrated with episodes from the teaching experiments, and we argue that
inferentialism offers a fruitful resource for thinking about learning and teaching more
generally.
4.1. CHALLENGES
Challenge 1. With regard to inert knowledge we argue that if students ostensibly learn
‘knowledge’ in the absence of reasoning with representations that make sense to them, then
this knowledge is likely to become inert and hard to transfer to new situations. In the case of
the arithmetic mean, the ‘inferential distance’ between the mean as a calculation and the
28
mean as a measure of center useful for comparing groups is very large for most middleschool students. This distance, consisting of several reasoning steps, makes it hard for
students who have mainly learned to calculate the mean to apply it as a measure of center (cf.
Zawojewski & Shaughenessy, 2000). Bakker (2004a) designed a series of instructional
activities that supported students to take non-trivial inferential steps with the help of
appropriate graphical representations. The third illustration given in this paper, in which
students used the mean to check if the GE fish grew twice as big as the normal fish, suggests
that middle school students can indeed use the mean inferentially, provided they have
experience with suitable learning activities and are guided by a competent teacher. In
inferentialist terms: students need to become aware of a larger web of reasons in order to use
the concept of mean in a useful way.
Challenge 2. In the teaching experiments referred to in this paper, we did not
atomistically address mean, median, mode, bar graph, and histogram, but tried to engage
students in meaningful contextual problems from the very start to evoke informal inferences.
Many (but not all) of these instructional activities involved comparing two data sets, which is
sometimes seen as the beginning of statistical inference (Watson & Moritz, 1999). This focus
on drawing data-based conclusions, along with one software tool being used for all
investigations (TinkerPlots), presumably brought more coherence than topic-by-topic
approaches. We should note that the sixth-graders already had some fragmented and
procedural knowledge of mean, median, and mode. It would be interesting to do a similar
teaching experiment with students who have not yet learned definitions or algorithms
belonging to these measures of center, and see if their inferential function can be really
privileged over their representational use.
29
Challenge 3. A holistic view on concepts does not imply that every instructional activity
should draw on all statistical key concepts. Careful sequencing, based on the inferential
function of concepts, is still necessary. Our activities started with simple situations of
comparing data sets of the same sample size and similar variation, delaying more
complicated issues such as comparing groups with different sizes (e.g., the fish problem). In
this way, awareness of different types of variation and sample size were only fostered in the
last periods of the teaching experiments.
4.2. INFERENTIALIST LESSONS
The first inferentialist lesson drawn, summarized as “understanding concepts primarily in
inferential terms,” was illustrated in Section 3 by showing how statistical concepts such as
arithmetic mean gained meaning in the context of comparing groups and with reference to
informal representations such as “clumps” and shapes. We showed how a carefully designed
instructional activity supported students, when drawing statistical inferences, to become more
aware of relevant aspects of data sets in terms of what experts might recognize as precursor
statistical notions of center, variation, distribution, and sampling. The analysis of that episode
also highlighted some of the inferential relationships between statistical concepts that might
be missed if the order of teaching is governed by the structure of the statistical discipline as
typically represented in statistics textbooks. In our view, inferentialism helps to underpin
theoretically such a choice.
To illustrate students’ informal inferential reasoning with statistical concepts we have
focused on those concepts in reasoning processes (Lesson 1). Those concepts were not taught
one by one (Lesson 2), and perhaps this was possible because the students had at least some,
30
though fragmented knowledge about mean, median, and mode—this is something to test
empirically in future research. Contextual problem situations were designed in which
students needed to become aware of aspects of all these concepts when making inferences
(Lesson 3). This meant that students did not end the teaching experiment with fully
developed statistical concepts or definitions of statistical measures such as standard deviation
or interquartile range. But in this classroom practice, statistical features of data sets and
graphs such as clumps, mean, and variation became topics of attention, tools to reason with
and talk about. In other words, students developed awareness of many aspects of data sets
that are in our experience typically ignored in cases where statistics education takes a
representationalist approach.
In response to the questions raised in Section 2, about the relationship between concepts
and reasoning, inferentialism pleas for educators understanding concepts in terms of
inference—in the broad philosophical meaning, including implicit and tacit reasoning steps.
As an illustration of why we think it is important to draw attention to inference in this
Brandomian sense, we cite Wild’s (2006, p. 10) observation on variation and distribution:
There are aspects of statistics that are so basic to the way we think in the subject that no one
abstracts, enunciates and examines them. We encountered this phenomenon frequently in
conducting the research for Wild and Pfannkuch (1999). It is not a problem for the statistical
practice of professionals since they have long since been successfully encultured into these ways
of thinking. It may well, however, be a root cause of some of the problems we face in statistics
education. “Variation” was one of these unenunciated givens until quite recently and still is for
many communities of statisticians. “Distribution” is another fundamental given of statistical
reasoning.
31
What Wild in our view points to is the fact that in statistics viewed as a practice (in
Brandom’s sense of the term), understanding and the use of reasoning is built on a preexisting experience of the inferential relations which allowed concepts to be acquired in the
first place. Competent users need not be conscious of the inferential relations out of which
their expertise was formed. Indeed to be conscious of them would probably sabotage their
competence, but when the same expert is teaching statistics s/he must be aware that what they
are taking for granted is something their pupils have still to learn. The term “unenunciated”
stresses the mostly implicit and unarticulated inferential domain of communication (web of
reasons) that is unproblematic for knowers of the field but difficult for students who are still
to be initiated into the practice of statistics as a discipline. Given a professional’s reasoning
fluency with the key representations in the field, one might easily fall prey to privileging the
explicit and visible representations and even taking them to be the disciplinary knowledge.
We therefore see inferentialism as a valuable reminder to keep in mind the existence of an
implicit inferential domain (a web of reasons) supporting and sustaining the meaning of
representations. It is in this sense that Brandom’s phrase “privileging inference over
representation” should in our view be understood, and that inferentialism can assist to
underpin theoretically a focus on informal statistical inference.
4.3. STATISTICAL INFERENCE, CONCEPTS, AND CONTEXT
In the introduction we wrote that inferentialism helped us explore the relationships between
statistical inference, concepts, and context. So what does inferentialism in combination with
our illustrations have to say about the role of context and concepts in relation to informal
statistical inference? Within this special issue the notion of context refers to the non-
32
statistical domain that the inferences are about, for example, in the case here, the fish
farmer’s problem. It is fairly common in education to talk about statistics and context as
separate domains, and there can be good reasons to do so, for example to summarize
particular observations in classrooms. Students might be aware of statistical representations
and norms, but not those of fish farmers, or vice versa. It is also possible that students find
motivation in or have knowledge about the context that helps them to learn more about the
statistics.
However, from an inferentialist perspective a dichotomous distinction between statistics
and context is problematic. The distinction suggests that there is ‘text,’ in this case the
statistical representations, and ‘con-text’—what surrounds this text (cf. Roth, 1996).
However, as Brandom and Vygotsky make clear, a concept cannot be understood merely in
its representational form; its meaning is disclosed in a rich system of judgments about a
situation. These judgements involve commitments which are responsive to situations,
discourse, and artifacts, including the various representations typical of the discipline.
Judgments are constituted in and connected by inferential relations within a web of reasons
(Section 2.3).
It is for these reasons that Bakker et al. (2008) suggested the notion of ‘web of reasons’
as a more precise and non-dichotomous alternative to that of ‘context.’ A designer in our
view needs such a more precise notion, because s/he has to take into account all relevant
reasons involved in a educational problem. In the fish farmer case, we assumed students
knew very little about fish farming, but understood he wanted to have big fish to earn more
money and was therefore interested in their sizes. They also understood that he could not
catch all fish to test his conjecture. Apparently their awareness of these non-statistical reasons
33
was high enough to support their statistical inferences, even though their notions of center,
spread, and sample were still implicit. We assume that awareness of the fact that the farmer
could not catch all fish, even if not yet made explicit in some representational form,
eventually helped students to attribute meaning to the concept of sample. We suggest future
research should try to identify students’ awareness of the important reasons involved in
educational tasks and tease out how these help them develop rich statistical concepts. This
type of research faces methodological challenges because reasons and inferences can only be
inferred and reconstructed from perceivable behavior and representations.
4.4. FINAL POINTS
We have articulated an inferentialist view on knowing and proposed an inferentialist
approach to teaching ISI to underpin theoretically current research in this area. As illustrated
in this paper, such an approach to ISI means students are necessarily engaged in informal
reasoning with concepts-under-development with appropriate tools (cf., Radford, 2003).
However, as discussed by Sfard (2000), many people find the lack of rigor typically
accompanying such approaches problematic, and it would seem that this problem is not easily
solved. Papert (2003, p. ix), for example, observed a
tension that is most vividly represented for mathematics educators by the “MathWars” in which
reformists seeking greater engagement of students are pitched against mathematicians who
consider that engagement has been bought at the cost of rigor and denounce the reformist
curriculum as “fuzzy math.”
It would seem that approaches that favor active learning (especially if student terms such as
clumps and bumps are used) are at odds with the beliefs and values of mathematics educators
who want to maintain the rigor of their knowledge domain. At first sight, inferentialism
34
might seem to offer little more than other criticisms of representationalism found in
contemporary literature (e.g., Cobb et al., 1992; Sfard, 2008), however Brandom’s
inferentialism goes further and has more radical implications. In particular, it directs attention
to the knowledge domain, which has been sidelined in the efforts to undermine the
dominance of representationalism (Cobb et al., 1992). Lessons from inferentialism suggest
that greater attention be paid to students’ awareness in the very first instance when this is
understood in terms of the background knowledge domain in which their learning takes
place. Both awareness and the knowledge domain have in our view been neglected in
constructivist and sociocultural theory. To speak of a knowledge domain such as statistics is
not to refer to anything fixed (a commodity) but rather to a field within which concepts are
connected to each other by virtue of the uses made of them. If we are not attentive to this
further dimension of awareness then something will be lost in the efforts to foster active and
engaged students. This paper has illustrated what statistics education could look like if we
privilege holism over atomism and inference over representation in an inferentialist approach.
ACKNOWLEDGMENTS
We are grateful to the teacher and her two classes for taking part in the teaching experiments,
Teri Hedges and family for their hospitality, and Eena Khalil for letting us use her fish data
cards. This paper benefited from a long history of anonymous reviewers’ thoughtful
comments. We thank editors Iddo Gal, Katie Makar, and Dani Ben-Zvi as well as esteemed
colleagues Cliff Konold, Michael Hoffmann, Anna Sfard, Paul Drijvers, Anneke Verschut,
Sanne Akkerman, and Dor Abrahamson for their helpful suggestions. The TinkerPlots
35
research was funded by the National Science Foundation under grant number ESI-9818946.
Opinions expressed are those of the authors and not necessarily those of NSF.
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