lessons from inferentialism for
statistics education
Arthur Bakker
Freudenthal Institute, Utrecht University
jan derry
Institute of Education, University of London
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 ofISI. We suggest that inferentialismcan serve asa valuable theoretical resource for reform efforts thatadvocate ISI. To illustrate what it means to privilege an inferentialistapproach 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
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 disposalmight 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 paperis 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 alsoon 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 compatible with Vygotsky’s ideas, due to the influence of Hegel.Although several of the lessons we draw from Brandomresonate 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 wereflect on the main issues brought up in the paper (Section 4).
- 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 threerelated challengesthat are persistent in education in general and manifest themselves in statistics education in specific ways.
- The challenge to avoid inert knowledge—knowledge students have learned to reproduce but cannot use effectively (Bereiter & Scardamalia, 1985; Whitehead, 1929).
- The challenge to avoid atomisticapproaches found in many textbooks and to fostercoherence from a student perspective.
- The challenge of sequencing topics in alternative approaches which aim for coherence from a student perspective.
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. Manystatistics textbooks addresslists 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 t-test 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 suchgraphical representations that students can draw data-based conclusions. In line with the recent literatureon ISI we takeISI to be central in statistics education and treat key concepts such as 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 itin 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 statistical concepts or their informal precursors in making inferencesprior 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 isa 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 abroader 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 usethe 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, formulas, models, written and spoken statements etc.—anything that stands in place of something else (i.e. anything that is re-presented).
- 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 lessonwe 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 summarizewhat can be characterized as representationalismand 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
Representationalismrefers to the position that representations are the basic theoretical constructof 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 whereconceptual contentis 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:
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 clarifieshis 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 non-standard contexts.
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 thoughpeople 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 reasonsstill 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 made only a few explicit. A knowledgeable person is responsive to arich web of reasons in such contexts. Some reasons might be statistical (the fish farmer might even want to do a t-test 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, statisticalconcepts such as mean, variation, distribution, and sample should be understood in terms of their role in reasoning, i.e. in terms of the commitments entailed by their use.As we argue here, learning statistical conceptsin 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.