Proper and paradigmatic metonymy as a lensfor characterizing student conceptions of distributionsand sampling

Jennifer Noll & Stacey Hancock

# Springer Science+Business Media Dordrecht 2014

Abstract This research investigates what students’ use of statistical language can tell us abouttheir conceptions of distribution and sampling in relation to informal inference. Prior researchdocuments students’ challenges in understanding ideas of distribution and sampling as toolsfor making informal statistical inferences. We know that these ideas are complex and difficultfor students, but little is known about the ways in which students’ language mediates theirstatistical problem-solving activities within the realm of distribution, sampling, and informalinference. This study uses examples from semi-structured interviews with eleven undergrad-uate students from two universities. Interview tasks focused on (1) distinctions betweendistributions of populations, samples, and sample statistics; (2) properties of sampling distri-butions; and (3) how to use sampling distributions to make informal inferences. Analysisfocused on students’ use of metonymy (i.e., the substitution of the name of an attribute oradjunct for that of the thing or idea meant). We observed two particular metonymies. The firstwas a paradigmatic metonymy in which students applied the properties of the normaldistribution to all distributions. Second, we observed a proper metonymy in which studentstalked about sampling distributions as compilations of many samples. The potential impact ofthese metonymies on students’ ability to solve problems and the implications for teaching arediscussed.

Keywords Introductory statistics . Statistical reasoning .Distributions . Samples and sampling .

Informal statistical inference . Informal inferential reasoning .Metonymy

Educ Stud MathDOI 10.1007/s10649-014-9547-1

This paper is submitted for the Special Issue of ESM on “Statistical Reasoning”

Research in this paper conducted while at Clark University, Worcester, MA.

J. Noll (*)Portland State University, 724 SW Harrison, Neuberger Hall 334, Portland, OR 97201, USAe-mail: noll@pdx.eduURL: http://www.mth.pdx.edu/~noll/

S. HancockUniversity of California, Irvine, Bren Hall 2019, Irvine, CA 92697, USAe-mail: stacey.hancock@uci.eduURL: http://www.ics.uci.edu/~staceyah/

1 Introduction

Data based, informal inferential thinking is a fundamental skill for participation in a global,technological, and data-driven society. Yet there are complex and subtle concepts that connectideas of distribution and sampling to informal statistical inference, and understanding of theseconcepts requires mastery of statistical language and terminology. As human beings attempt tograsp new abstract concepts, they use language as a bridge between the familiar and concreteto the new and abstract. Vygotsky (1934/1986) wrote that a “word without meaning is anempty sound” and “word meaning is both thought and human speech” (p. 6). Lakoff andJohnson (1980) argue that insight into a human’s conceptual system can be gleaned by lookingat language because “communication is based on the same conceptual system that we use inthinking and acting” (p. 3). Thought and language are so intimately intertwined that educatorssimply cannot ignore the role of word meaning in the process of thinking if they want to makemeaningful headway into student cognition. Thus, the goal of this research is to investigatewhat students’ use of statistical language can tell us about their conceptions of distributions,sampling, and the connection of these ideas to informal statistical inference.

Lakoff and Johnson (1980) argue that the way in which people think and act is fundamen-tally metonymic and metaphoric in nature.

Metonymies and metaphors are both relationships

“between two entities, A and B, where B represents A. However, in the case ofmetaphor, A and B are located in two different conceptual domains, rather than beingpart of the same conceptual domain as in the metonymic relationship” (Zandieh &Knapp, 2006, p. 3).

For example, “the suits onWall Street” is a metonymy; suits stand in for the people who work onWall Street or, more generally, financial corporations. In this example, suits are part of the conceptualdomain for Wall Street and those who work there. Whereas, “life is a journey” is a metaphor; all theentailments of a journey aremapped to the life of a person, and these are distinct conceptual domains.

Although the constructs of metonymy andmetaphor have received somewhat limited attentionin mathematics education, and virtually no attention in statistics education, cognitive scientistsargue that metonymies and metaphors are more than simply figures of speech; rather, they framethe way we make sense of the world, express complex meaning, construct alternatives, andorganize systematic concepts and by doing so engage in disciplinary discourse (see, for example,Abrahamson, Gutiérrez, & Baddorf, 2012; Cortazzi & Jin, 1999; Kovecses, 2002; Lakoff &Johnson, 1980; Presmeg, 1998; Zandieh & Knapp, 2006). Lakoff and Núñez (2000) argue thatmetonymy is a “cognitive mechanism that permits general proofs in mathematics” (p. 75).

A review of the literature reveals two distinct kinds of metonymy: “metonymy proper” (orsynecdoche) and “paradigmatic metonymy” (Presmeg, 1992, 1998; Zandieh & Knapp, 2006).Presmeg (1998) defines a proper metonymy as using “one element or salient attribute of a classto stand for another element or the whole class” (p. 26). Zandieh and Knapp use the example“I’ve got a new set of wheels” (p. 2) for proper metonymy, where one part of the car, thewheels, stands in for the whole car. A paradigmatic metonymy is when “a part is a prototypefor the whole category” (Zandieh & Knapp, 2006, p. 3). For example, letting a particulartriangle represent all classes of triangles is a paradigmatic metonymy.

Presmeg (1992) provides compelling evidence that students use mathematical images inmetonymic and metaphoric ways, which are key components in their mathematical develop-ment. If metonymies and metaphors go beyond figures of speech into the realm of supportingand structuring human cognition, and mathematical images are key tools in this process, then it

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is plausible that the cognitive constructs, metonymy and metaphor, are key pieces in the puzzleto better understand students’ statistical development.

There are differing views as to how the constructs of metonymy and metaphor might beleveraged in educational research with respect to providing insight into theories of learning (e.g.,Lakoff & Núñez, 2000; Radford, 2013). The work of Lakoff and Núñez as well as that ofPresmeg (1992, 1998) suggests that people may use metaphors and metonymies at times withintention but perhaps more often without intention. However, other researchers (Abrahamsonet al., 2012) suggest that when a person does not use a metaphor or metonymy intentionally, thenit is not truly a metaphor or metonymy, and they use the term “inadvertent metonymy.”We findthat there is nothing inherent in the definition of these words that suggests the person speakingneed be aware he/she is using a metaphor or metonymy and follow in the steps of Lakoff andNúñez as well as that of Presmeg. Regardless of one’s perspective on the intentionality of themetaphor or metonymy on the part of the user, all of these researchers initially characterizedstudents’ language as being metaphorical or metonymic in nature as a lens (a tool on the part ofthe researcher) from which to better understand human cognition. This seems particularlyplausible when thinking about concepts of distribution, sampling, and the relationship to informalinference. These concepts require abstraction from a specific case (e.g., individual data point orindividual sample) to the general (e.g., distribution of a sample or distribution of sample statistics,and the variability inherent in a distribution of data). In addition, via instruction, these conceptstypically have strong graphical images associated with them (e.g., normal distribution). Thus, weconjecture that teachers and students are likely to use metonymies and metaphors as a vehicle forthe construction of new knowledge from prior knowledge when learning statistical concepts.

Abrahamson et al. (2012) propose a similar conjecture about the use of metaphors by teachersand students inmathematics, suggesting that when students need tomake sense of new ideas, theydo so by determining what the new ideas are analogous to. “In this sense-making process,students recall familiar phenomena. To communicate their insight—how they are seeing thephenomena—students use metaphor” (p. 58). In many respects, our research complements that ofAbrahamson et al., though we focus on metonymy rather than metaphor and its use in statisticsrather thanmathematics courses. The innovation in our research is in the application ofmetonymyto study statistics students’ thinking about distributions, sampling, and informal inference.

We focus on two research questions:

1. What examples of metonymies arise in statistics within the context of distribution,sampling, and informal inference?

2. What insights into students’ developing conceptions of distribution can we gain by usingmetonymy as a lens?

We focus specifically on students’ use of metonymy because in the context of our research, wedid not see metaphors arise when students were communicating about distribution, sampling, andinformal inference. This is not to say that students do not use metaphors in these contexts, onlythat we did not observe this in our data. Students’ language was metonymic in nature because thedifferent parts of their speech remained within the same conceptual structure rather than mappingbetween different structures. To illustrate this point, consider a student who describes a symmetricdistribution as a hill or a speed bump. This description would be metaphoric in nature because thestudent is mapping properties of speed bumps to properties of symmetric distributions, and theseideas lie in different conceptual domains. A student who describes a sampling distribution as acompilation of many samples is using a metonymic description where lots of samples (a part)stand in for the whole concept of sampling distribution. The latter types of descriptions wereobserved in student excerpts and thus form the basis for the research presented in this paper.

Proper and paradigmatic metonymy as a lens

In the sections that follow, we synthesize the background literature on the use of metonymyas a cognitive construct in mathematics education; then we conjecture how such a constructcould be relevant within the context of distribution, sampling, and informal inference instatistics education. We situate this discussion within the background literature on studentthinking about these statistical contexts. We provide an analysis of metonymies that appearedin student interviews and discuss important implications for student learning and teachingpractice.

2 Background

Presmeg (1992) argues that any time a student uses a diagram to reason mathematically this ismetonymic usage of the diagram or image. She suggests two ways in which students’metonymic usages of diagrams may be problematic. First, a student may have difficulty inthe form of “non-recognition of a figure when it does not conform to a prototype” (p. 600). Forexample, a student may not recognize a triangle as a triangle if it does not contain a right angleor may potentially apply properties of a right triangle when asked to reason about triangles ingeneral. Second, a student may experience difficulty if they introduce “extraneous propertieswhich are present in the figure but not necessarily in the general case” (p. 600).

The epistemological point of view of Bakker and Hoffmann (2005) that “abstract conceptsbecome visible in signs and in their use in mathematical activity” (italics in original, p. 335)shares similarities with Presmeg’s point of view that students’ activity is mediated by theirmetonymic usages of diagrams. Additionally, Abrahamson and Wilensky (2007) write, “theinherent ambiguity of diagrams requires that a learner adopt a conventional way of seeing thediagram so as to participate in a social practice that utilizes these diagrams unambiguously” (p.5). In their work, they attempt to create bridging tools to help students reconcile ambiguity.The epistemological viewpoints of these different researchers all consider how students cometo understand abstract concepts through their interaction with diagrams. That is, students’ useof diagrams can tell us something important about their thinking and reasoning. In addition,these researchers describe how the use of specific diagrams can either support or hinderstudents’ reasoning about general cases of the concept of which the diagram is a representative.

We hypothesize that the two problematic metonymic usages of diagrams described byPresmeg could also pose significant challenges in developing students’ statistical thinkingaround distributions, sampling, and the relationship of these concepts to informal inference.Yet, these challenges need not be inherently problematic. Consider the work of Abrahamsonet al. (2012). In their study, they observed that students’ metaphors were powerful insupporting their progress from nonnormative to normative mathematical views, but thistransformation did not happen in isolation, rather through careful attention and effort on thepart of the interviewer so as to leverage students’ metaphors during instruction.

We begin our focus on students’ use of metonymies in statistics with distributions. Wild(2006) argues that we look at distributions of data to learn “widely applicable lessons,” but the“lessons are not, we believe, to be found in the individual data points themselves, but inpatterns discernible in the dataset as a whole” (p. 11). He states, “the distinction that underliesdiscussions of empirical versus theoretical distributions is between the variation we see in ourdata and a potential model for the process that gives rise to that variation” (emphasis in theoriginal, p. 13). This means that students need to be able to distinguish between the specificdistribution generated from their data and a general probability model, a (theoretical) distri-bution of a random variable likely to have generated the empirical distribution at hand. IfPresmeg’s hypothesis (student reasoning with diagrams is inherently metonymic in nature) is

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true, then a logical consequence is that students may naturally apply properties of a particulardistribution to all distributions or apply extraneous properties of a particular distribution (likethe outliers or a student’s particular data value) to reason about patterns in the distribution athand. In fact, statistics educators have already noted students’ initial tendencies to reason aboutindividual data values, rather than consider aggregate features of a data set (Bakker &Gravemeijer, 2004; Ben-Zvi & Arcavi, 2001; Konold & Higgins, 2002). Prodromou andPratt (2006), however, argue that students need to “co-ordinate” data centric and modelingperspectives and that students’ initial focus on the local should not be seen as a misconceptionof distribution as much as a developing conception.

Wild (2006) argues that “[w]e make distributional assumptions in order to come up withmethods of drawing inferences from data” (emphasis in the original, p. 14). Thus, distributionsare an important conceptual building block for understanding ideas of informal inference,which lead to formal statistical inference, a key focus for most college-level introductorystatistics courses. Yet, research suggests there are substantial gaps in students’ informal andformal understanding of statistical inference (e.g., Chance, delMas, & Garfield, 2004;Pfannkuch, 2005; Zieffler et al., 2008). These researchers have documented that students havedifficulty making the connection between probability models and statistical inference; thisconnection is based on the idea of sampling distributions. Sampling distributions describewhat we would expect to see if we could collect many samples (and thus sample statistics)from a population, enabling us to compare a statistic from an empirical sample with atheoretical distribution of sample statistics. Researchers have documented students’ myriaddifficulties with the concept, indicating that students often confuse distributions of sampleswith distributions of sample statistics, apply the properties of the population distribution to thesampling distribution, experience difficulty making sense of the variability of sample meansand how this knowledge can be useful in statistical inference, and aggregate multiple samplesinto one large sample rather than look at a distribution of statistics from multiple samples(Chance et al., 2004; Lipson, 2003; Saldanha & Thompson, 2002, 2007).

The student difficulties and developmental stages documented in this prior body of researchcould be recast through the lens of metonymy: focusing on the specific case at hand (individualdata points, a local view with potentially extraneous properties) in place of the general (globalview), applying properties of a particular distribution to all distributions, or aggregatingmultiple samples into one large sample. Also, research that documents potentially “problem-atic” metonymies that may hinder a student’s ability to make informal inferences or look foroverall patterns in the data could be used to help teachers understand these potential usages andhow to leverage them in helpful ways in instruction so that indeed the metonymy is notproblematic at all, but rather a natural step in the learning process. In order to investigate therole of metonymy in student thinking about distributions, sampling, and the relationships toinformal inference, a holistic consideration of the conceptual domain of distribution is neededso as to understand how particular parts of that domain might be used to stand in for the whole.In order to develop a conceptual domain of distribution, we investigated how statisticians andstatistics educators discuss this concept.

Chance and Rossman (2006) define the distribution of a variable as “the possible outcomesof the variable and how often each outcome occurs relative to the other outcomes” (p. 8). Wild(2006) discusses “distribution as a lens through which statisticians look at variation in data” (p.10). Bakker and Gravemeijer (2004) suggest that distribution is “an organizing structure or aconceptual entity” (p. 149) for looking at data. Table 1 shows Bakker and Gravemeijer’sstructural mapping between data and distribution, where an “upward perspective leads to afrequency distribution of a data set” (p. 149) and a downward perspective uses a probabilitydistribution to model data.

Proper and paradigmatic metonymy as a lens

The characterizations and models of distribution offered by these researchers provided anexcellent starting point but were insufficient for proposing a conceptual domain of distributionuseful for conceptualizing metonymies. A further synthesis of the research literature ondistribution (see Bakker & Gravemeijer, 2004; Chance et al., 2004; Gil & Ben-Zvi, 2011;Pfannkuch, 2011; Pfannkuch & Reading, 2006; Prodromou & Pratt, 2006; Wild, 2006)suggests that knowledge of distribution includes (1) individual data values and outliers; (2)data as an aggregate collection; (3) summary features of the distribution—shape, spread, andcenter; (4) the role of variability in the outcomes of a distribution; (5) different types ofdistributions—population, sample, and sampling distribution; (6) empirical versus theoreticaldistributions; (7) probability; (8) coordinating exploratory data analysis perspectives withmodeling perspectives; (9) different graphical representations of distributions of data and theaffordances or impediments of different representations; and (10) the role of context instudents’ development of informal inferential reasoning about distributions of data. The ideasof distribution mentioned here include both conceptual underpinnings of a distribution (e.g.,summary features of data and connections between probability and statistics) as well asoperational issues, such as how best to display and summarize the data as well as makeinferences from the data (Pfannkuch & Reading, 2006, p. 4). In order to construct a frameworkof metonymy that could apply to distribution, we synthesized all the different components(representations, attributes, classes of distributions, inferential interpretations) for the conceptof the distribution based on a thorough review of the literature (see Table 2). Each componentrepresents a potential part of the entire conceptual structure and thus potential metonymies forthinking of the whole of distribution.

The framework in Table 2 highlights different conceptual underpinnings for conceiving ofdistributions of data. For example, a distribution could be represented graphically (with ahistogram, dot plot, box and whisker plot, etc.), as a table showing the relative frequency ofoutcomes, or a distribution can be characterized by its global features (e.g., mean, range,shape). Inferences can be made from a distribution by looking at signals in noise (in themanner of Konold & Pollatsek, 2002) or via a simulated sampling distribution from which onecan compare an observed statistic to a distribution of statistics and calculate an empirical pvalue. It may be that students develop understanding of distribution in one component of thetable through their understanding of distribution in another component.

The concepts of distribution and sampling and ideas of inference are intimately intertwined,and the ideas of sampling and inference are inherently part of the broader conceptual domainof distribution. Thus, the conceptual domain of distribution outlined in Table 2 provided astarting point for our investigation of the role of metonymy in student thinking aboutdistributions, sampling, and the connections to informal statistical inference. The specificresearch questions that guided our analysis were the following: (1) What types of metonymiesarise in statistics within the context of distribution, sampling, and informal inference? (2) What

Table 1 The mapping of Bakker and Gravemeijer (2004) between data and distribution (p. 148)

Distribution (conceptual entity)

Center Spread Density Skewness

Mean, median, midrange,… Range, standard deviation,interquartile range,…

(Relative) frequency,majority, quartiles

Position majorityof data

Data (individual values)

J. Noll, S. Hancock

insights into students’ developing conceptions of distribution can we gain by using metonymyas a lens?

3 Methods

3.1 Institutions and participants

Data were collected at a medium-sized private comprehensive college (site 1) and at a smallprivate liberal arts-based research university (site 2), with primarily traditional students (ages18–20). Data collection at site 1 was conducted in an undergraduate introductory statisticsclass, primarily containing students who were completing a minor in mathematics. There were20 students enrolled in the course. Data collected from site 2 was conducted in an undergrad-uate introductory statistics class for environmental science and geography students. There were25 students enrolled in the course, primarily first year and second year students. All students atboth sites consented to be participants and allowed their work to be collected as data. Bothcourses were nontraditional in the following sense: they used textbooks aimed to lead studentsthrough activities where they “construct” statistical ideas. Students at both sites also usedtechnology to run statistical simulations as a way of constructing sampling distributions andanswering inference questions. The statistical activities and assessments (which includedcomputer-generated simulations) were geared toward providing students opportunities toinvestigate distributions of data. The activities and assessments provided an environmentwhere students could practice statistics by engaging in statistical conversations about distri-butions of data, and this also provided us a context from which we could investigateimplications of student language for their thinking.

3.2 Surveys and interviews

Students completed a written assessment toward the beginning and end of the courseconsisting of nine multiple-choice and short-answer questions. The problems, which involvedquestions about repeated sampling and sampling distributions, were borrowed from ARTIST(Assessment Resource Tools for Improving Statistical Thinking; see https://app.gen.umn.edu/artist/) scales and the CAOS (Comprehensive Assessment of Outcomes in a first Statisticscourse; see https://app.gen.umn.edu/artist/caos.html) test. Toward the end of the semester,students were invited to participate in an hour-long interview. Out of the ten volunteers at site

Table 2 Conceptual domain for thinking about distributions of data

Components of the conceptual domain of distribution

Representations Attributes Classes of distributions Inferential interpretations

Graphs (box plots,histograms, scatterplots, dot plots, piecharts, etc.), tables,and spreadsheetformulas

Local features (individualdata points, outliers)and global features(measures of center,spread, shape)

Empirical versustheoretical, types ofdistributions (normal,uniform, binomial, chi-square, etc.), distributionof sample, distributionof population, and sam-pling distribution

Signals in noise, p value,and empirical rule;variability within andbetween samples; andconnections betweenprobability models andstatistics

Proper and paradigmatic metonymy as a lens

1, six were selected (three females, three males) who represented a range of abilities (thestudents had received midterm grades of A through C−). There were 13 volunteers at site 2; tenof these volunteers had participated in the class activities. Of these ten, six students (fivefemales, one male) were randomly selected to participate in interviews. The interviews fromsite 2 also represented a similar range of abilities as at site 1. At each site, the interviews werevideotaped and transcribed. One interview from site 2 was lost due to an inaudible recording.The analysis in this paper is based on these eleven interviews; students’ written assessmentresponses will be used in future analyses.

The interviews included four structured questions that were designed to investigatestudents’ understanding of the distinctions between different types of distributions andstatistical “unusualness” (i.e., how sampling distributions could inform students aboutinformal inference questions phrased in terms of identifying an unusual sample andwhy). The first question asked students to describe what a distribution is and todistinguish between the distribution of a population, the distribution of a sample, anda sampling distribution (i.e., the distribution of sample statistics). Two of the ques-tions were adapted from the assessment. In one of these questions, students wereprovided with a histogram of a sample of life spans of 200 tires as well as ahistogram of an empirical sampling distribution of average tire life spans for 200samples of size 50. Students were asked to interpret the data in each of the graphs,discuss the centers and variability of each graph, and make (informal) inferencesabout new samples of tire life spans (of size 50 and size 100) based on the previouslycollected tire data. In the second question adapted from the assessment, students weregiven the graph of a population and four graphs of possible empirical samplingdistributions for 1,000 samples of size 1, size 4, and size 50. Students were askedto discuss properties of sampling distributions in terms of shape, center, and variabil-ity (this task is not addressed in this paper). Finally, students were given one questionthat was similar to one of the in-class activities. In this question, students were givenhistorical data on platypuses’ nest sizes and they were also given the average nest sizefor a sample of ten newly collected platypus nests. Students were asked to determineif the new sample seemed unusual and to explain why or why not. The exactinterview protocol is shown in the Appendix.

3.3 Analysis

The analysis of students’ responses to the interview tasks went through multiplerounds of open coding (Strauss & Corbin, 1990) in order to identify themes instudents’ language. In particular, there were four iterative cycles in our analysis.Cycle 1 consisted of initial coding where four statistics educators read through theinterview transcripts making initial observations and reflections with an eye towardthe language students used. In this round of coding, summary descriptions of studentreasoning were created independently by each of the four coders and salient excerptsfrom the interviews were identified. Cycle 2 consisted of the several meetings withfour statistics educators sharing observations and summaries from the analysis of cycle1, looking for places of agreement and disagreement. Wherever there was disagree-ment, the researchers discussed, debated, refined, and eventually arrived at consensusfor new codes and characterizations of metaphorical and metonymic themes within thedata. Cycles 1 and 2 were initially focused on potential metaphors in student excerpts,but through the iterative cycles of independent coding, followed by discussion ofcodes, the team came to consensus that student language was more metonymic in

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nature because the relationships students described were all contained within theconceptual domain of distribution (rather than language that connected two differentconceptual domains). Thus, we limit our discussion to metonymies that seemed to beparticularly relevant in student discussions about distributions, sampling, and therelationship to informal statistical inference. In cycles 3 and 4 of our analysis, wereturned to the data with a focus on metonymic structures embedded in studentexcerpts and how these metonymies may play an important role in student develop-ment. Table 3 shows the two types of metonymies (paradigmatic and proper) andprovides some examples of such metonymies in the context of distribution that wereobserved in student interviews.

It is important to note that the metonymy framework we developed (Table 4) wasnot in place a priori. Through our iterations of independent data analysis, followed bygroup discussions of the metonymic themes we observed in the data, we created andrefined our theory of metonymy for distribution. In addition to reviewing interviewtranscripts looking for themes in student language, the research team coordinated theconceptual domain of distribution (as outlined in Table 2) with knowledge from theliterature review as to which aspects of distribution are difficult for students andrecast these themes through the lens of metonymy. Using this framework, we ob-served two uses of metonymy as the students in this study reasoned about distribu-tions. First, we observed a paradigmatic metonymy in which students used the normaldistribution (or its properties) as a prototype for any distribution of data. Second, weobserved a proper metonymy, in which students used a collection of samples to standin for a collection of sample statistics when discussing sampling distributions. Thesetwo uses of metonymy are described further in section 4.

4 Results

This section discusses the ways in which we observed the use of metonymy in students’discourse around concepts of distribution and perhaps more importantly, the consequences ofmetonymies on thinking about connections between distribution, sampling, and inference.Through our iterative cycles of analysis (as described in section 3), the metonymic frameworkfor distribution as outlined in Table 4 was developed. Hence, this framework also serves as aprimary result of our work. This section is divided into four parts. First, the framework ispresented and described. Second, normal distribution as a paradigmatic metonymy for distri-bution is illustrated through student excerpts. Third, a compilation of many samples as a proper

Table 3 Examples of metonymies for distribution observed in student interviews

Metonymy Description Examples

Paradigmatic One part of the concept is aprototype for the whole concept

1. Normal distribution as a prototype forall distributions

2. Dot plots as a diagram/representation of alldistributions

Proper One part of the concept stands infor the whole concept

1. Compilation of samples stands in for samplingdistribution

2. Mean stands in for summarizing entire distributionof data

Proper and paradigmatic metonymy as a lens

metonymy for sampling distributions is illustrated through student excerpts. Finally, a sum-mary overview of the findings is provided.

4.1 Metonymic framework

Table 4 provides the resulting framework for the two metonymies observed in the analysis ofthe interview data. Under the description column of the table, there are indicators that served asevidence for a student’s use of a particular metonymy. The third column of the table providespossible ways these metonymies may help support or hinder student learning.

Table 4 Metonymy framework for distribution

Metonymy Description Potential consequences of metonymy

Normal distributionas aparadigmaticmetonymy

When a student uses the normal distributionas a prototype for all distributions

Support—students are developing ideasabout sample size, area under the curve,and probability. Students also learn tomake inferences based on area under thecurve where a statistic falls relative to thetails

Indicators: Hinders—applying normal distribution todistributions that are not normal leads toerroneous conclusions

1. Applying properties of normaldistributions to distributions that are notnormal or it is unknown if they arenormal

2. Applying the empirical rule todistributions that do not meet the normalassumption

3. Applying ideas of the central limittheorem to a distribution of a singlesample rather than sampling distributionscreated by samples of a given size

Proper metonymyfor samplingdistribution

A part being substituted for thewhole—referring to a samplingdistribution as a compilation of manysamples

Support—allows students to express acomplicated idea in fewer words andprovides the idea of repeated samplingand how more data can provide betterinformation from which to makeinferences. Compiling all samples intoone large sample seems to provide anatural inclination toward seeing the lawof large numbers and could be leveragedin instruction to get at this concept moreformally

Indicators: Hinders—if students aggregate all repeatedsamples into one large sample, then theymiss how variability among samplestatistics and the distribution of samplestatistics support statistical inference

1. Defining a sampling distribution as acollection of samples (rather than acollection of sample statistics)

2. Viewing the results of many repeatedsamples as one large sample

3. Inclination to make inferences about“unusualness” of a sample statistic basedon frequency information of sample data(rather than the distribution of samplestatistics)

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4.2 Paradigmatic metonymy: normal distribution is a prototype for all distributions

Normal distribution as a paradigmatic metonymy was observed when students had to reasonabout distributions, as in the tire task1. In this task, students were given two distributions ofdata displayed in histograms. The first was a sample of 200 tires, displaying the life spans ofeach tire (histogram A in Fig. 1), and the second was an empirical sampling distribution of 200trials of size 50, displaying the mean life spans of each sample of 50 tires (histogram B inFig. 1).

Students were asked (1) to estimate the mean life span of the brand of tires, (2) to discusswhether or not they used graph A or B or both to make their estimate, and (3) to explain what apopulation standard deviation of 5 years would mean to them. In responding to the standarddeviation question, we observed that 7 of the 11 interviewees applied the empirical rule, aproperty of normal distributions, arguing that about 68 % of the tires would have life spanswithin ±5 years of the mean and 95 % of the tires would have life spans within ±10 years of themean. Four of these seven students, unprompted, did add the stipulation that the populationneeded to be unimodal and symmetric. For example, John (site 1) said, “Well if I know thepopulation is unimodal and symmetric it would be 68 % of all tires would fall within 1standard deviation.” However, students had graph A in front of them, which is clearly notunimodal and symmetric, yet they continued to apply the empirical rule. In a few instances, theinterviewer prompted students, asking them if the unimodal and symmetric assumptions theywere making were true and what they would do if these assumptions were not valid. In all ofthese cases, students did not know what they would say about the standard deviation and thedistribution of the life span of tires if they could not apply the empirical rule.

Ted (site 1) is the best example of a student articulating his desire to apply properties of thenormal distribution to all distributions and his realization that he would not know what to do ifhe could not make these assumptions.

Ted (Site 1): I could pretend that this [graph A] is part of a sampling distribution, eventhough I know it’s not. …So, if I assume that the population fits a normal curve, then Icould use the empirical rule which tells me that 95 % of the data is going to be withintwo standard deviations of the center.

There were numerous times during the interview where this student suggested that he couldalways pretend he had a sampling distribution that would fit a normal curve and this wouldallow him to use the empirical rule. This suggests that the student wanted to use a normaldistribution as a prototype for any distribution of data that he needed to analyze and hencemake inferences from. As the next excerpt shows, he realized he would be stuck if he could notapply properties of normal distributions:

Ted (Site 1): I just realized I’m kind of making assumptions this entire time.... but if I didassume that it was skewed to the right I’d be totally lost.

Although it may be reasonable to argue the percentages of the empirical rule hold for thedata presented in histogram A, students did not use the graph to provide them with thisinformation and most of these students did not have a sense of what to do when normaldistribution assumptions could not be satisfied. The approach used by these students issuggestive of using properties of normal distributions as prototypes for all distributions of data.

1 See Appendix for a complete list of interview tasks.

Proper and paradigmatic metonymy as a lens

4.3 Proper metonymy: sampling distribution is a distribution of many samples

As indicated in Table 4, referring to sampling distributions as a collection of many samples ordifferences from sample to sample (rather than as a distribution of sample statistics) is anexample of a proper metonymy. Students used this proper metonymy in two different ways.First, they used this proper metonymy when defining sampling distribution. Second, they usedthis proper metonymy when using sampling distributions to make inferences.

4.3.1 Definitions of sampling distributions

Asking students to define sampling distribution is one way to gain insight into how they thinkabout this concept and whether metaphoric or metonymic themes arise in their definitions. Ofthe 11 students who participated in interviews, four students defined sampling distributionsusing some form of the metonymic shortening defined above, one student could not give adefinition of sampling distribution, and the remaining six students gave a complete definitionof sampling distribution as a distribution of sample statistics in their initial descriptions. Forexample, Jane (site 2) stated that a sampling distribution is “the difference from sample tosample.” This student never articulated the distribution as consisting of the statistics from eachsample. Two of the four students who used this proper metonymy did, later on, articulateexactly what they meant by a compilation of many samples and in both cases, thestudents referenced sample averages from the collection of samples. For example,Melissa (site 2) stated, “A sampling distribution is when you have a bunch ofdifferent samples and then distribute it over a graph,” but then later in the discussionsaid, “So you have a bunch of means.”

Fig. 1 Distributions in the tire task

J. Noll, S. Hancock

4.3.2 Using sampling distributions to make inferences

Asking students to define a sampling distribution is only one way to investigate how studentsmay use metonymies. Indeed, if metonymies frame the way we think, thinking about samplingdistributions as collections of many samples has important consequences about the way onemight apply sampling distributions to problems of statistical inference (see Table 4). It is notclear if a student who makes such a metonymic shortening of the definition literally thinksabout sampling distributions as a collection of many samples that can be compiled into onelarge sample or thinks about a collection of statistics from those samples. In addition, a studentmay articulate the definition of sampling distribution with perfect clarity but still showevidence of thinking about sampling distributions as a way to collect one very large samplewhen applying their knowledge of sampling distributions to statistical inference. In order to geta better sense of whether students’ metonymic shortenings about sampling distributions wereindicative of a nonnormative (or developing) view of sampling distribution, we investigatedhow they talked about sampling distributions when making inferences in the tire and platypustasks.

In the tire task, students were asked to compare a new sample of tires to the informationpresented in the histograms. They were told that a new sample of 50 tires resulted in a mean of6 years and were asked if this provided evidence that the new tires had a longer average lifespan than the old tires (the mean of the sample of old tires was close to 5 years, see Fig. 1). Ifone literally considers sampling distributions as a collection or compilation of lots of samples,then a natural consequence of such a view would be to combine the samples into one largesample when making an inference.

Jane’s (site 2) definition of sampling distribution as “the difference from sample to sample”was metonymic. Karen (site 2) was unable to provide a definition of sampling distributionwhen asked directly. Both of these students tended toward compiling lots of samples into onelarge sample, a consequence of the metonymy of sampling distribution as a compilation of lotsof samples, when making an inference about the new sample mean of 6 years.

Jane (Site 2): Well in this case they tested more tires [points to histogram B] and like sixwas not as frequent, but in histogram A they tested less tires but it [6] was frequent. So Ithink there’s some evidence but I’m not sure if it’s strong evidence.Karen (Site 2): Because it’s more, more data technically because you have two hundredpeople looking at fifty tires each so it’s ten thousand tires that are being looked at insteadof two hundred…. This one [Graph A] only uses two hundred tires and this one [GraphB] uses ten thousand tires and like averaged it. So I feel like histogram B would indicatemore if it was unusual.

Both of these students suggest that there are more data/tires when considering graph Bbecause they mentally compiled all the samples of size 50 into one large sample. Karen andJane’s reasoning that graph B has more data and is therefore a better source of information is areasonable and logical response to the question. From these students’ perspectives, thisresponse is sufficient to answer the problem; thus, without appropriate instructional interven-tion, the students may not develop the perspectives of sampling distributions that statisticseducators would like to see2.

Melissa (site 2) used the same metonymic shortening in her initial definition of samplingdistribution as Jane and Karen (site 2), but in contrast, she went on later to add “sample means”to clarify her description. In addition, her method for determining the unusualness of a new

2 This point will be elaborated on in further detail in the discussion section.

Proper and paradigmatic metonymy as a lens

sample was to compare where the new mean fell in relation to a distribution of means asopposed to a large distribution of individual data points. For example, the following excerptshows how she used histogram B, the sampling distribution, to decide whether a mean lifespan of 6 years in the new sample of 50 tires is unusual.

Melissa (Site 2): They did the same exact study [referring to Histogram B], but withdifferent tires. So since it’s exactly the same they have a higher mean [pointing to wherethe new sample mean falls in the distribution of statistics for Histogram B] so that wouldmean that their tires are lasting longer.

Susan (Site 1) described each element of a sampling distribution as “one distribution from asample (pause), one average”. Later on when describing properties of sampling distributionsshe drops the use of average and uses samples to stand in for sample average.

Susan: Because it’s a sampling distribution, so we can use the central limit theorem toknow that it’s going to be more symmetric as we get more samples - or larger samples.

Like Melissa (site 2), Susan uses this metonymic shortening and is not hindered by it. Infact, it appears to allow her to speak more directly about inference ideas without having toexplain the entire concept of sampling distribution. We do not mean to suggest here that Susanand Melissa are explicitly aware that they are using a metonymic shortening, only that theyseem to naturally shorten their language about sampling distributions in this way, which allowsthem to move forward in their inference discussions.

A final example from the interviews displays evidence of a student being in transition.When Ted (site 1) defined sampling distribution, he gave a complete and correct definition:“It’s like a distribution, just instead of each individual data point being an individual data point,it’s the sample statistic for one sample.” Yet, when he needed to use a sampling distribution tomake an inference in the platypus task3, he appeared to oscillate back and forth between seeinga need to compare the new sample mean with a distribution of means versus comparing thenew sample mean with a distribution of a single, large sample.

Ted (Site 1): I’m just trying to decide whether my sample is going to be a single platypusnest or not. I’m leaning towards yes…. They did 10 samples [referring to the researchersin the problem] so I’ll do 10 too. So, actually, I’ll just do 1,000. … Essentially I’m justrunning a lot of trials and then we can look at all of the response values.…If I looked at1,000 platypus nests, what’s the probability that the nest is going to have 4 eggs.

In this first excerpt, the student is describing how he will determine if a mean of 4 eggs isunusual. He uses the language of creating a sampling distribution by collecting lots of samples(trials), but his samples are of size 1 and he discusses finding the frequency of nests with foureggs. However, as the student continues to consider the problem, he contemplates why he hasnot used the “10” or the average number of eggs as discussed in the problem.

Ted (Site 1): I feel like I’m not accounting for the 10 platypus nests….I mean, that’s howmany they looked at. So I guess they kind of have their own little sampling distributiongoing on, don’t they? Unless I look at each nest as an observational unit and the numberof nests they look at as the sample, which would be a different…Actually, I could do thatif I upped the number of observations to 10. So now every trial that I run I have to switchit to an average. Now I’m taking ten tests and finding the average.

3 See Appendix for details on the platypus task.

J. Noll, S. Hancock

Here, the student makes a switch from language that implies running lots of trials of size 1(to get one large sample) to language that implies tracking the average number of eggs forevery sample of ten nests. However, he ultimately ends up going back to consideringindividual nest sizes.

Ted (Site 1): Now I’m not looking at an actual observation anymore. I’m looking at anaverage.Interviewer: Is that bad?Ted: It’s bad…because there’s never going to be an average of 7 because 35 % of themare going to be 1.

Ted appears to understand what a sampling distribution is, but oscillates between makinginferences about unusualness based on one large sample (many trials of size one) versuslooking at the distribution of statistics and the variability among those statistics. Again, both ofTed’s approaches are logical, and he is clearly attempting to make sense of the statisticalsituation. He appears to be grappling with the distinction between collecting one large sampleand looking at the sample mean versus collecting lots of smaller samples and looking at themeans of each of those samples. He also appears to be considering what each of theseapproaches buys him in terms of his ability to make inferences. Ted’s two approaches highlightwhat may be a natural struggle for many students in the progression toward normative views ofstatistical inference.

5 Discussion and conclusions

Our goal was to investigate metonymy as a lens for viewing student reasoning aboutdistributions, sampling, and the relationships to informal statistical inference. Analysis ofstudent work on the tasks described in this study reveals possible affordances and challengesof metonymies in the teaching and learning of statistics. This section highlights theseaffordances and challenges as well as providing directions for future research in whichmetonymy serves as a lens for viewing student reasoning.

5.1 Affordances of metonymies

Mathematics education researchers have suggested metonymies and metaphors can serve aspowerful mediators, allowing students to build mathematical objects for discourse that enablethem to solve new problems (e.g., Presmeg, 1998; Zandieh & Knapp, 2006). In statistics,graphs provide one example of a concrete object to make the focus in statistical conversationsand may support student movement from the specific (local features or particular empiricaldistributions) to the general (global features or theoretical distributions). From this perspective,metonymic usages of graphs may be important for developing student understanding of aholistic conception of distribution. Though we only used histograms in the interview questionsfor this study, there exist multiple types of representations of a data set or model (e.g., tables,formulas, and other types of plots such as dotplots, bar charts, scatterplots). We plan to useother representations, including theoretical distributions, in future research on student use ofmetonymy.

Sampling distributions are complex concepts. Using samples as a stand in for samplestatistics has some advantages. If a student implicitly understands the ideas left out by ashortening of the definition, then it suggests that they can use sampling distributions as objectsfrom which to make inferences about an observed sample mean. This is similar to advantages

Proper and paradigmatic metonymy as a lens

of metonymies as discussed by Zandieh and Knapp (2006). In these instances, metonymiesfree students from having to articulate every detail about a process they already understand andallow students to use these concepts in the construction of new knowledge about theconnections between sampling distributions and informal inference. In addition, unintention-ally using metonymy to compile repeated samples into one large distribution from which tomake inferences has advantages for finding a natural starting place from which to introduce thelaw of large numbers.

5.2 Challenges of metonymies

Despite the possible advantages of metonymy, there are some metonymies that could poten-tially hinder students’ progress in understanding distributions, especially when these meton-ymies are not leveraged in instruction or, worse still, go unnoticed by the instructor. In ourstudy, we observed two metonymies that students employed which led them to use certainstatistical ideas in nonnormative ways. First, we observed that several students used normaldistributions and their properties as a prototype for all distributions (paradigmatic metonymyfor normal distributions). This specific imagery could potentially hinder a student’s ability tomake progress in certain statistical tasks where the data set may not warrant the normalityassumption. Students need to understand the data as well as the context within which the dataare set so that they do not attempt to apply properties for a particular distribution, such as thenormal distribution, to any distribution.

Such paradigmatic metonymies have more important implications for teaching and suggestthat teachers may need to do a better job of using other types of distributions in classroomdiscussions so as to avoid the natural inclination of students who might apply properties ofnormal distributions to all distributions. Recall the example of Presmeg (1998) of a studentwho tries to apply properties of a right triangle to all triangles. The application of properties ofa normal distribution to any distribution is analogous to Presmeg’s triangle example. Inreflecting on the content of most introductory statistics courses, a large percentage of classtime is devoted to normal distributions and their particular properties. Cobb (2007) makes acompelling argument that we spend too much time focused on the normal distribution in classand that students are likely to see it as the center of the statistical universe (p. 4). This issue wascertainly the case with many of the students in our study.

In addition, when teachers introduce the empirical rule for normal distributions or thecentral limit theorem to discuss the effects on the distribution of sample means as sample sizeincreases, teachers may often focus on specific important details such as the relationshipbetween probability for particular outcomes and the area under the curve (of the normaldistribution) and the role of sample size (on the shape of the distribution of sample means).That is to say, teachers may naturally shorten sentences (and/or add metaphors—bell-shaped)saying, “As sample size increases, the distribution (of sample means) becomes bell shaped” asa way to help simplify ideas for students. However, what is implied in these sentences includeseverything in parentheses whether or not teachers actually say it. These shortened phrases aremetonymic in nature and have important implications for how students approach distributionsof data and the conclusions they may draw.

A sampling distribution is a distribution of sample statistics. A proper metonymy forsampling distributions could be that sampling distributions are collections of samples, differ-ences from sample to sample, or compiling a bunch of samples. In these cases, samples maystand in for the particular statistic calculated from each of the samples. This proper metonymyfor sampling distributions (sampling distributions as the collection of many samples) may haveadvantages, as noted above, when students implicitly understand that the distribution is made

J. Noll, S. Hancock

up of statistics. However, for at least two students in this study, the metonymy was “problem-atic” because they literally compiled many samples into one large sample in order to makeinferences. The literature review articulated this issue: Recall the work of Saldanha andThompson (2007). They engaged students in a teaching experiment that required students tomake comparisons between different samples and between different sampling distributions.They noted that one student did not use the distribution of sample statistics collected fromrepeated samples; rather, she aggregated the repeated samples into one large sample and usedthe one large sample to make an inference. Yet, this student was able to articulate thedistinction between the observational units in the graph of a sample versus the graph of asampling distribution. Our research observed similar findings but looks at student reasoningthrough the lens of metonymy.

Zandieh and Knapp (2006) described students’ metonymic “misstatements” regarding theconcept of derivative as a potential natural shortening of a phrase in conversation or as apotential incorrect understanding of the concept. Because metonymic statements structure ourthinking, analyzing students’ metonymic “misstatements” may be useful for understandinghow they think about particular concepts. Metonymic “misstatements” defining samplingdistributions as a collection of samples may indicate that students are just using short handto discuss sampling distribution, or it may be indicative of a different conceptual view of whata sampling distribution is or how it is useful in inference.

Without thinking about sampling distributions as a collection of statistics, one has no wayto compare the likelihood of getting different values of those statistics. The distribution ofstatistics is needed to compare the variability among statistics and the potential “unusualness”of a statistic from the observed sample. Thus, on the one hand, the metonymic shortening ofthe definition of sampling distribution as a collection of samples may be problematic if ithinders students’ ability to see the connection between a distribution of statistics and theconcept of inference. On the other hand, statistics educators should not necessarily consider astudent’s desire to compile lots of samples into one large sample (rather than look at adistribution of sample statistics) as a misconception. Rather, this perspective is not withoutlogic and may represent a natural developmental progression. For example, if the questionbeing asked focuses on the center of the distribution, then there is nothing wrong withcompiling all the samples into one large sample and computing the mean for that sample. Infact, students’metonymic shortenings of sampling distributions could be useful for introducingthe law of large numbers and discussions around the impact of sample size. Also, with properleveraging of these students’ metonymic statements, instructors could find ways to build fromthese statements in classroom discussions so that students can develop normative views ofsampling distributions (for examples with metaphor in probability see Abrahamson et al.,2012). Could the interviewer have used students’ metonymies in these examples to engage thestudent in a way that would have supported the construction of a sampling distribution toanswer the statistical question? Further research needs to be done to see how prevalent such aperspective is among introductory statistics students and how educators might capitalize onthis perspective in instruction.

5.3 Concluding remarks

It is worth noting that some of the “problematic” uses of metonymy mentioned here are not sodifferent from the kinds of misunderstandings that could take place in conversational Englishwhen metonymies are used and one of the participants in the conversation is not a nativeEnglish speaker. For example, when someone says, “I have a new set of wheels,” we do notthink that this statement is incorrect (although technically it is) because we would implicitly

Proper and paradigmatic metonymy as a lens

know that the person is referring to having a new car. Yet, if there are nonnative Englishspeakers participating in the conversation, the metonymy may confuse them. They mayunderstand the sentence literally or take away a different understanding altogether. This samesituation arises in the classroom. In statistics classrooms, we have a teacher (hopefully fluent inthe language of statistics) and students (new learners of the language of statistics). Whenteachers focus on particular distributions or images of those distributions, students may takethose images as prototypical. If teachers naturally shorten sentences about important ideas ofdistributions, sampling and inference, students may not see what is implicit within thosestatements.

Teachers may also use metonymic devices, without realizing it, when introducingstatistical concepts. Teachers of statistics realize that the elements being left out of ametonymic statement are naturally implied, but their students may not. For instance,the metonymic shortening of the definition of sampling distribution from a “distribu-tion of sample statistics” to a “distribution of samples” leaves out key informationthat our interest is in the statistic from each of the samples. The metonymies andmetaphors that teachers use when introducing new statistical terminology and conceptsare likely to impact student learning and is an area for future research. Teachers mustbe precise in their definitions and be aware of metonymies they may be using duringinstruction and how metonymies may support or hinder student development ofparticular ideas.

Other researchers have discussed the importance of learning to reason aboutdiagrams through the use of semiotic frameworks (e.g., Bakker & Hoffmann, 2005)or through bridging tools (e.g., Abrahamson & Wilensky, 2007). Abrahamson et al.(2012) studied three students’ use of metaphors when working on probability prob-lems and how, via instructor discussions with these students regarding their meta-phors, students’ metaphors helped them transform their understanding of the mathe-matical situation. They focused on metaphor, whereas our work focused on metony-my. Our work offers an additional lens with which to consider how students usediagrams as well as how they understand particular ideas of distribution. LikeAbrahamson et al., our work also reports on a relatively small sample size, and thushas no statistical power. In addition, our interviews were not designed as tutorials forteaching and therefore the interviewers made no attempt to follow up with studentmetonymies. Further work on ways of leveraging student metonymies in instructionshould be included in future studies.

We all use metaphor and metonymy regularly in every day conversations, and most of thetime we are unaware that we are doing so. Despite the limitations, this work is importantbecause the ways in which students use metonymies and metaphors to construct statisticalobjects of discourse and how that process enables them to solve statistical problems is criticallyimportant for improving our understanding of student cognition and how it develops. Theresearch presented here starts a conversation about metonymy as a lens for studyingstudent thinking about ideas of distribution. In addition, knowing how student under-standing of statistical concepts may develop as a result of metonymic cognitivestructures has important implications for teaching statistics. Further study in this areashould investigate metonymies and metaphors in statistics, how these constructs canenhance or hinder students’ statistical development, and the ways teachers may beable to build off of these ideas in instruction.

Acknowledgments The authors wish to thank Aaron Weinberg and Sean Simpson who gave us excellentfeedback and suggestions on various versions of this paper and participated in all parts of the data analysis.

J. Noll, S. Hancock

Appendix

Interview protocol

Definitions

1. Could you describe to me what a distribution is in statistics?

a. Could you give an example?2. Could you describe what a “distribution of the population” is?

a. Could you give me an example?3. Could you describe what a “distribution of a sample” is?

a. Could you give me an example?b. How is it similar to or different from a distribution of the population?

4. Could you describe to me what a sampling distribution is?

a. Could you give an example?b. How is it similar to or different from a distribution of a sample?

Tire task

A new type of tire is tested to see how long its tread lasts. In one study, a researcher sampled200 tires, measured how long each one lasts, and graphed her results in Histogram A (below).In a separate study, 200 people each looked at 50 tires; they each found the average life of their50 tires and then they graphed their 50-tire averages. Histogram B shows the sample averagesof the 200 people.

Proper and paradigmatic metonymy as a lens

Ask the student the following questions.

1. What do the numbers on the horizontal axes represent?

a. Students may ask if they need to give a single response that covers both graphs; tellthem that they can describe them in different ways if they’d like.

2. What do the numbers on the vertical axes represent?3. Which of the distributions has more variability? Please explain your reasoning.4. Approximately what is the mean life span of the tires?

a. How did you get your answer?b. If not enough detail is given:

i. Could you tell me how you were thinking about the situation?c. Did you use both histograms equally, or did you focus more on one of them?

i. Why did you choose to focus on that histogram?ii. Is there a way you could have also used the other histogram?

5. [Use a highlighter to shade in the bar above “6 years” in each graph]

a. What does the shaded bar in Histogram A represent?b. What does the shaded bar in Histogram B represent?

6. If students struggle with either of the previous two questions, ask:

a. If Histogram A/B were a dotplot, what would each dot represent?b. If the student is confused, draw horizontal lines across some of the bars to make it

look like a dotplot (except with a stack of boxes instead of dots)c. If they can’t think of an answer, suggest that they reread the problem description.

7. Let’s assume that the standard deviation of the population was 5 years.

a. Could you explain what this tells you about the life of the tires?b. Could you tell me how you were thinking about the situation?c. What do you think the standard deviation of the data in Histogram A might be?

Explain your reasoning.8. If you know that the population’s standard deviation is 5 years, how does the standard

deviation in Histogram B compare to the population’s standard deviation?

a. Should it be larger, smaller, about the same, or is there no way to tell?b. Do you think it makes sense for Histogram B to have a smaller standard deviation

than Histogram A?c. Why do you think it makes sense?d. Could you tell me how you were thinking about the situation?

9. Michelin recently came out with a new type of tire that they claim has a longer life thanthe tires that are represented in these graphs. Let’s say a group of researchers goes outand buys 50 of these new Michelin tires and finds that they had an average tread life of6 years. Do you think this is evidence that Michelin’s tires really last longer?

a. Did you use Histogram A or B (or neither) to answer this question?b. Could you tell me how you were thinking about the situation?

J. Noll, S. Hancock

c. [Continue to probe here to unearth the student’s conception of sampling distributions.However, answering this question essentially involves performing a hypothesis test;make sure you don’t get too caught up in the logic of hypothesis testing.

10. Let’s say in the previous question the researchers had purchased 100 Michelin tiresinstead of 50 tires.

a. Would this change your answer from the previous question?

i. If so, how?ii. Explain your reasoning

b. Let’s imagine that Histogram B didn’t exist, and you only had Histogram A and your100-tires sample.

i. How might you decide whether the new type of tire lasts longer than other tires?ii. Did you use a sampling distribution to help you think about the situation? If so,

please describe the sampling distribution and how you used it.

Platypus task

Put the following description on the table in front of the student and read the description aloud:Ecologists are studying the breeding habits of the Australian River platypus.

These platypuses lay between one and seven eggs each year. Based on historicalresearch, the ecologists estimate that the number of eggs occurs with the followingfrequencies:

Eggs Relative frequency

1 35 % of all nests

2 25 %

3 15 %

4 10 %

5 5 %

6 5 %

7 5 %

[“Read” the first three lines of the table: “The ecologists think that 35 % of all Platypusnests have 1 egg in them,” etc.]

One day, the ecologists go out on one stretch of the Brisbane River, and they find tenplatypus nests, with an average nest size of 4 eggs. They want to determine whether this isunusual. How could you construct a sampling distribution to help them determine this?

If the students are stuck, try giving one of the following prompts:

a. How could you make a graph of the information in this table? What do the values in thegraph represent?

b. What does the 10 % for four eggs mean to you?c. Imagine taking a sample of ten platypus nests from the distribution defined in the table.

What would you expect the sample mean to be?d. How might you use simulation to help you address this question?

Proper and paradigmatic metonymy as a lens

a. How would you set up the simulation so that it modeled the percentages in the table?b. What would a sample consist of?c. What would a trial consist of?d. What would you measure in each trial?e. How many trials would you conduct to convince yourself of your answer?

e. How could you construct a graph to display how sample means would vary? What do thevalues in this graph represent?

f. What do we mean by “unusual”?

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