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Statistical Modeling to Promote Students’ Aggregate View of Data

in the Context of Informal Statistical Inference

Keren Aridor and Dani Ben-Zvi

LINKS I-CORE, University of Haifa, Israel

Abstract

Helping students develop an aggregate view of data is a key challenge in statistics education.

It has been suggested that modeling pedagogy can address this challenge (Lehrer &

Schauble, 2004). In this paper we present a case study – part of a UK-Israel research project

– that aims to examine how students’ reasoning about modeling of a real phenomenon can

support the emergence of aggregate view of data, in the context of making informal statistical

inferences. We focus on the emergent reasoning of two fifth-graders (aged 10) involved in

statistical data analysis and modeling activities using TinkerPlots2. We describe the students’

articulations of aggregate view of data as they: 1) explore a small sample; 2) plan and

construct a model that represents the investigated phenomenon and make predictions about

‘some wider universe’; and 3) generate random samples from this model to examine its

representativeness. This paper aims to contribute to the study of models that young students

can understand and use to develop their aggregate view of data.

Keywords: Exploratory data analysis, informal statistical inference, aggregate view of data,

statistical model, statistical modeling.

Introduction

One of the core aspects of statistical reasoning is handling data from an aggregate point of

view (Hancock, Kaput, & Goldsmith, 1992), namely, viewing data as an entity with emergent

properties, such as shape, center and spread (Konold, Higgins, Russell, & Khalil, 2014).

Young students tend to see data as individual cases and measurement values as inseparable

from an object or person measured. Students who cannot develop a notion of an organizing

structure with which they can see the whole instead of just the elements, miss the essential

point of doing statistics, which is predicting properties of aggregates (Bakker & Hoffmann,

2005). Therefore, developing students’ aggregate view of data is a key challenge in statistics

education (Bakker, Biehler, & Konold, 2004). It has been suggested that placing statistical

modeling at the heart of statistics learning can address this challenge by supporting students’

search for patterns in data and account for variation in these patterns (Pfannkuch & Wild,

2004). In this paper, we closely study this assertion. This case study is part of a UK-Israel

research projecti (Ainley, Aridor, Ben-Zvi, Manor, & Pratt, 2013) that demonstrates how

fifth-graders’ modeling of an authentic phenomenon using TinkerPlots2 (TP2, Konold &

Miller, 2011) can support the emergence of aggregate view of data. We focus on the ways

they shifted between local and aggregate views of data while reasoning about models and

their context in the context of making informal statistical inference.

Literature review

Informal Statistical Inference (ISI) is a relatively new theoretical construct and pedagogical

approach aiming at deepening learners’ understanding of statistical inference in relation to

other key statistical ideas (Garfield & Ben-Zvi, 2008). ISI is based on making generalizations

beyond the given data, expressing uncertainty using a probabilistic language and using data

as evidence for those generalizations. The main goal of teaching ISI is to deepen the

understanding of the purpose and the gain that can be driven from the data and its

interpretations (Makar & Rubin, 2009). The reasoning process leading to making ISIs is

Informal Inferential Reasoning (IIR, Ben-Zvi, Gil, & Apel, 2007; Makar, Bakker, & Ben-Zvi,

2011). IIR is a cognitive activity engaged in formulating generalizations (e.g., conclusions,

predictions) from random samples of data using various statistical tools, while considering

and articulating evidence and uncertainty. IIR involves reasoning with several key statistical

ideas such as: sample size, sampling variability, controlling for bias, uncertainty and

properties of data aggregates (Rubin, Hammerman, & Konold, 2006).

Aggregate view of data. Statistical thinking is developed from a partial or local view of data

toward a global view of data (Konold et al., 2014), and the ability to flexibly shift between

these views (Ben-Zvi & Arcavi, 2001a). Such reasoning is called aggregate view of data, or

aggregate reasoning. Konold et al. (2014) defined a hierarchy of three other perspectives for

viewing data that are taken by students and are encapsulated by aggregate reasoning: 1) data

as pointers to the context of the source of the data, without referring to the data itself (there is

no fundamental unit). Data cases are served as reminders to the larger event from which it

came (e.g., refereeing to events that happened during the data collection and are not

necessarily seen in the data); 2) data as case values that provide information about the value

of some attribute for each individual case. Individual cases are perceived as the fundamental

unit for analysis and focusing on their characteristics (e.g., focusing on extreme values); 3)

data as classifiers, that give information about the frequency of cases with a particular

attribute value. Such cases are perceived as a unit with similar properties (e.g., the mode of

the data). The way that the data is viewed depends on the purpose of the data collection, the

context of the problem, and on the questions that are asked, and influences the way the data is

handled, e.g., the research questions, data representations, interpretation from data and

inference (Konold et al., 2014).

When viewing data as an aggregate, a data set is considered as an entity, or as a group,

with emergent properties, which are different from the properties of the individual cases

themselves (Friel, 2007). The notion of distribution as an organizing conceptual structure is

supported by aggregate reasoning (Bakker & Gravemeijer, 2004) that allows concentration on

the distribution’s emergent features such as: the general shape, how spread out the cases are,

and where the cases tend to be concentrated within the distribution (Konold et al., 2014).

With categorical data one might describe frequencies using percentage or quantitative

descriptors (e.g., “most”, “majority”), or with numeric data, one might relate to properties

such as measures of center (i.e., mean, median), of shape (e.g., symmetry, skewness), of

density (actual or relative frequency, majority, quartiles) and of spread (e.g., outliers, range,

interquartile range, standard deviation) (Friel, 2007; Cobb, 1999). Two important aggregate

properties are: 1) distinctions between signal and noise; and 2) recognition and diagnosis of

various types and sources of variability (e.g., variability due to measurement error, natural

variability, sampling variability) (Rubin et al., 2006). A pedagogical approach placing

modeling in the center of data exploration can support the emergence of aggregate view of

data (Lehrer & Schauble, 2004).

Model and modeling. Models are analogies in which objects and relations in the model

system are used as stand-ins to those in the real world by means of representations, laws, and

structures of reasoning (Lehrer & Schauble, 2010). Modeling is a process of forming a model

on the basis of key theoretical aspects and data in a particular discipline, and evaluating and

improving it to include theoretical ideas or new findings (Lesh, Carmona, & Post, 2002).

Modeling is considered a form of explanation that is characteristic – even defining – of

science. Model-based reasoning entails deliberately turning attention away from the

investigated phenomenon to construct a model (Lehrer & Schauble, 2010). A modeling

approach puts the modeling process (along with learning about the nature and the purposes of

models) in the center of the learning process (Schwartz & White, 2005).

Mathematical models are abstract constructs that focus on structural characteristics or

on a general pattern that is common to several systems (Lesh & Harel, 2003). Mathematical

models are used in statistics to represent a general pattern of the data (Moore, 1990). Model

and modeling are essential components of statistical reasoning (Wild & Pfannkuch, 1999).

The practice of statistics is a form of modeling, as the development of models of data,

variability and chance are paving the way of the statistical investigation (Wild & Pfannkuch,

1999; Lehrer, Kim, Ayers, & Wilson, 2014). A statistical model have an important role in the

foundations of statistical thinking, and reasoning with models is considered as a general as

well as specific statistical type of thinking. The former relates, for example, to statistical

conceptions of the situation that influence how we collect data about the system and analyze

it, and the latter relates, for example, to measuring and modeling variability for the purpose of

prediction, explanation, or control (Wild & Pfannkuch, 1999; Garfield & Ben-Zvi, 2008).

The main usages of statistical models are: 1) selection, design and usage of a suitable

model to simulate data that will address a research question. For example, by using a tool that

generates random data (e.g., dice), or by simulating a distribution of a population, based on a

sample of real data, that can be used to make inferences while examining statistical concepts,

such as representativeness of the sample (as in this case study); and 2) adaptation of a

statistical model to databases in order to explain or describe the variation in the investigated

population, for example, adjusting a linear model to data that describes a relationship

between variables (Garfield & Ben-Zvi, 2008).

A modeling pedagogical approach that views data as a model of a situation in the real

world (Hancock et al., ) can serve as a bridge between data and probability (Konold &

Kazak, 2008) by providing multiple affordances to learn about random samples and sampling

from an investigated population, consider key statistical ideas emanating from the study of a

hypothetical model of this population, and examine the connections between these elements

(Manor, Ben-Zvi, & Aridor, 2013). For example, modeling a random behavior (the

randomization test) might provide an opportunity to experience and reflect upon probabilistic

situations. It allows to mimic such behavior in a real world system, to answer questions about

that system and to predict future outcomes. Modeling random behavior underpins the

quantification of uncertainty using statistical inference techniques such as confidence

intervals and significance testing (Arnold, Budgett, & Pfannkuch, 2013). A modeling

pedagogical approach can support learners in coordinating their understanding of particular

cases with an evolving notion of data as an aggregate of cases (Lehrer & Schauble, 2004),

among others, by the need to summarize data in multiple ways depending on its nature

(Pfannkuch & Wild, 2004).

In this study we consider a model to be an analogy which simplifies a real

phenomenon and describes some of the connections and relations among its components. A

model can emerge through an observation of the real phenomenon, while selecting and

focusing on features that are relevant to a specific purpose, for which it was constructed. A

model might be abstract (conceptual) or concrete (e.g., graph, table, dice, TP2 sampler). The

abstract model can represent a real world system and the conjectures about it in order to

describe, explain, predict, and elaborate on its behavior (Wild & Pfannkuch, 1999; Lehrer &

Schauble, 2010). A concrete model can serve as a tool representing a process, such as a

production of the population, its key components or properties through prediction or by

sampling or as a tool that supports the emergence of informal ideas (Garfield & Ben-Zvi,

2008).

We assume that each step of the statistical investigation entails a process of

emergence, development, refinement or verification of a conceptual or concrete model,

according to a certain need or purpose. This process is related to an emergent ability to view

globally a real phenomenon in the context world. This view might entail conflicts between

context and data, which can support the development of aggregate view of data. In this case

study, a conceptual model, followed by concrete models, were developed by a pair of

students in an attempt to describe a real phenomenon, make predictions about it, and

“produce” its population. We focus on the emergence of these models in relation to students’

views of data.

Method

The research question. In this paper we focus on the question: How did the modeling of an

investigated phenomenon play a part in promoting (or hindering) the emergence of students’

aggregate views of data? In order to address this question, we use data of a pair of fifth grade

students (aged 10) as they participated in the Dalmatians Task – an authentic inquiry of

exploration, prediction and explanation of statistical modeling in the context of ISI. This case

study is a part of a UK-Israel collaboration (2012-2014) aimed at developing and studying a

modeling approach for teaching and learning statistics by integrating the benefits of

Exploratory Data Analysis (EDA) and Active Graphing (AG) (Ainley & Pratt, 2014; Ainley

et al., 2013).

The setting and participants. The participants are Iddo and Yael, a pair of academically

successful and articulate ten year-olds (grade 5) from two Israeli public schools. The students

had no previous formal experience in statistics or TinkerPlots. Both students learned earlier

this year in school how to calculate the arithmetic mean. The students spent three hours on

the Dalmatians Task.

Data collection and analysis. Two researchers introduced the task and the tools and

frequently asked the students to clarify their reasoning. The students’ investigations were

fully videotaped using Camtasia and an additional video camera to capture both their

computer screen, discussions and actions. The videos were carefully observed, transcribed,

translated from Hebrew to English, and annotated for further analysis of the relationship

between modeling and the development of students’ aggregate view of data. We used the

interpretative microgenetic method (Siegler, 2006) to analyze the data. It is a qualitative

detailed analysis of the transcripts that take into account verbal, gestural, and symbolic

actions within the situations in which they occurred. Interpretations were discussed by UK

and Israeli researchers until a consensus was reached. Episodes were selected to illustrate the

students’ development of aggregate view of data using modeling in the context of ISI.

Differences between Hebrew and English connotations of words were discussed extensively.

The Dalmatians Task. The children were asked to plan a model that would “produce”

realistic Dalmatians of different sizes in order to create a theme park of the 101 Dalmatians

movie. The learning trajectory (Table 1) was designed to encourage the students to reason

with key statistical ideas (such as, models, distribution, center and variability (signal and

noise), sample and sampling), express uncertainty, and develop aggregate view of data.

Table 1. The Dalmatians Task learning trajectory.

Content

Min.

a) Introduce and discuss the task

Learn about the task and make conjectures about the dog population

Ideas \ concepts

Natural variability, reality vs. simulation

5

b) Collect data

Measure two real Labradors (we had no Dalmatians at hand) and discuss their properties and the

relations between them

Ideas \ concepts

Natural variability, relations between attributes

18

c) Discuss and analyse a realistic data of five Dalmatians

We provided data of five Dalmatians’ spot color, height, tail length, body length, and leg length (Fig.

1). The students were asked to make conjectures, test them and search for relations between the

attributes using TP2. The quantitative variables values were approximately simulated according to the

relations between body measures of real Dalmatians: body length is similar to height at shoulder, leg

length is between half and two thirds of height at shoulder, and tail length is a bit more than half body

length.

Figure 1: A realistic data of five Dalmatians in a TP2 table.

Ideas \ concepts

Variability, uncertainty, relations within and between attributes

48

d) Build a model (A ‘machine’)

Plan and build a model in TP2 (a ‘machine’) to produce realistic Dalmatians.

Ideas \ concepts

Distribution, range, center, variability, frequency, chance, reasonable

data

30

e) Draw random samples from the model (run the ‘machine’)

Draw random sample graphs and compare them to the realistic data graph.

Ideas \ concepts

Randomness, spread, chance, variability, population and sample, signal

and noise

89

f) Evaluate the model and improve it

Evaluate and improve the model according to the realistic data and expectations raised from the

context world

Ideas \ concepts

Uncertainty, randomness, spread, chance, variability, population and

sample, dependent and independent variable, signal and noise

10

Summary of findings

The following description is provided to serve as a background for the viewing and

discussion of the video segments at the conference.

a) Introduce and discuss the task. The researcher (first coauthor) introduced the task goal to

the students, and asked them: how could we generate realistic dogs that would be different

from each other? The students began to reason about the population and its characteristics,

considering variability in dog’s dimensions and temper.

b) Collect data. The students discussed first how to measure the real Labradors and then

measured them accordingly. Yael’s preliminary conjecture was that dogs’ body

measurements are related to age, but to their surprise, the older dog (two years old) was

smaller in all her measurements than the younger dog (9 months). When they analyzed the

Collection 1

Options

spots height tail_length body_length leg_length <new>

1

2

3

4

5

brown 41 23 40 22

black 37 23 37 18

black 26 13 27 14

black 30 19 30 16

black 30 15 31 17

collected data, they conducted a comparison between the two dogs’ measurements, as well as

between the measurements of each dog. They found that there was a variability between the

dogs (one is bigger than the other) and within the dogs (by the proportion between attributes).

They declared that: “dogs are very different from each other”.

c) Discuss and analyze a realistic data of five Dalmatians. After a short preview of the

software, the students started analyzing the data (Table ). They examined one variable at a

time in stacked dotplots, and relations between attributes in a scatterplot. Iddo saw a clear

relation between height and leg length. They examined their conjecture (Fig. 2a) and looked

mostly locally at the data, considering data as case-value by focusing on extreme values.

Figure 2a (Left): Relation between leg length and height. Figure 2b (Right): Relation between

tail length and height.

Although they noticed a pattern in the data that strengthened their conjecture, the

students were bothered by three cases - two, four and five (Figs. 1, 2a). Two of them (cases

four and five) had the same size of height (30 cm), and a similar size of leg length (16 and 17

cm) and the irregular case (case two) had a similar leg length (18 cm) but bigger height (37

cm). In attempt to make sense of this irregularity, the students searched for explanation in

other attributes. They concentrated on comparing the table’s columns and rows and looking at

graphs. For the rows, they found similarity between the values of the height and body length

for each case. The focus on cases four and five, led the students to isolate the tail length

attribute, arguing that this was the only attribute that distinguished between these cases and to

discover another interesting pair of cases - cases one and two (Figs. 1, 2b), that had the same

tail length, as the students noticed in the graph, but were differed in the other attributes, as the

students saw in the table. While wandering between the table and the graph, the students

revealed relations between attributes mostly by searching for similar values of two attributes

of each case. Iddo generalized these relations in a way that took variability into account and

said that a dog that is biggest in one attribute is relatively big in the other attributes.

Yael refined her method for looking at the table, and suggested another generalization

by referring to “the difference between attributes” of the same dog. She noticed that the

difference between attributes’ values for each specific dog were smaller than the difference

between attributes’ values of different dogs. Iddo used TP2 pen to draw a trend line and pairs

of parallel lines from some cases to the axes, to emphasize a proximate y=x linear relation

between height and body length (Fig. 3).

Yael had an idea for generating more dogs, but she didn’t express it clearly. She

decided to use a paper to describe a new discovery (Fig. 4) - a method to assess the strength

of a relation between attributes. She divided the four numerical attributes into two categories,

where the difference between the values inside each category are small, while the difference

between values from different categories are big. She referred to two attributes in the same

category as ‘closed’ (e.g., height and body length) and two attributes from a different

category as ‘open’ (e.g., body length and leg length).

Figure 3 (left): A trend line to emphasize the relationship between height and body

length. Figure 4 (middle): Yael’s discovery: types of relationships between attributes of a

phenomenon. Figure 5 (right): A model of the attribute height among Dalmatians in TP2.

d) Build a model (a ‘machine’) for one attribute. The students decided to model a relation

between two ‘closed’ attributes: height and body length. They suggested possible values for

the heights, while getting familiar with various TP modeling devices that were introduced to

them by the researcher. Yael referred to the height range in the table, and offered to slightly

increase it in the machine. They referred to the range and center of the height distribution,

considering the mean and the likelihood of a value to be close to the center.

When the students used the curve device, a conflict arose between them: Yael insisted

on drawing an approximately normal curve and Iddo tried to draw a bimodular curve. Iddo

explained his opinion by the different preferences people have for dog’s height, or by the

frequency of the height as he perceived it. Yael explained her motivation by the need to set

the heights according to the likelihood and chances of their occurrence. Although this

argument might suggest initial signs of aggregate view of data, the students didn’t look at the

distribution of the heights as a whole, and tried to set the frequencies of each value of the

heights in the model. They tried each of the TP2 devices: mixer, stacks, curve, pie and bars,

in order to search for the device that would allow them to do that easily, and decided to use

the bars that allowed them to set the percentage of each height and to set it easily by drawing

the cursor (Fig. 5).

e) Generate random samples from the model of height. The students took a random sample of

10 cases and tried to make sense of it. Iddo explained that the sampler chose values according

to the percentages given to it. They both said that if they sampled more dogs the “picture”

would be different and referred to the sample size as responsible for the absence of a certain

value that they set in the model.

f) Evaluate the model and improve it. The students added another bar device for the body

length. They set its range to be the same as the height’s range after examining the table, and

Yael stated that its “arrangement” did not have to be the same as the one they set for the

height. Once again they tried to model a normal distribution (Fig. 6). The children drew a

random sample of 10 from the model, and were surprised not to get a linear relation as they

expected. They tried to handle the noise in the data by editing the model, but neither changing

the frequencies of a certain range of the body length, nor reducing the range of it, helped. At

this point the researcher showed the students how to design dependency between two

attributes, and they separated the body length to five equal intervals, and set a uniform

distribution for each of them, explaining that they would change it later (Fig. 7). They were

more satisfied by the random sample generated from the new model as it provided them more

similar data to the original data table and a clear signal of the relationship between height and

body length. But this clear signal raised a new problem that the students acknowledged - the

lack of noise.

Figure 6 (left): A model of the relation between height and body length among Dalmatians in

TP2. Figure 7 (Right): An improvement of the model of the relation between height and body

length among Dalmatians in TP2.

A bit of a discussion. After measuring the two Labradors, the students began to explore the

Dalmatians’ data with the sense of a large variability in the population. The strong authentic

context encouraged them to search for trends and patterns in the data. It seems that the need

to model the investigated phenomenon encouraged the students to invent various types of

models in order to make sense of the data and to produce dogs. While searching for similarity

in the data and for explanations for irregularity, they developed methods to compare between

cases and attributes locally and then globally using the table. They verified their discoveries

in the data and refined them constantly. Their initial view toward data was local considering

data as case values. We suggest that the students’ focus on clusters of three cases elicited a

discussion about “rules”, that might be an expression of a rudimentary aggregate view. An

emergence of this initial reasoning was also seen when Yael suggested two categories to

assess the strength of a relation between attributes.

The modeling of a concrete model using TP2 seems to raise the need to take into

account the range, center and shape of distributions. This sense was tested when a conflict

was raised between the students about the attribute’s distribution. While Yael felt the need to

describe a smooth, normal and maybe theoretical distribution, Iddo searched for the sense of

irregularity in the model and tried to describe it in the distribution. The need to model a

dependency between attributes and to examine random samples, involved a refinement of the

model, along with reasoning about statistical ideas such as signal and noise, chances, sample

size, variability and uncertainty.

Expected Contributions

We hope that the results of this case study will contribute to the discussion about “aggregate

view of data” in relation to modeling approaches in IIR, as well as provide the grounds for

further research that will expand the existing knowledge about these issues.

Our selected video segments for SRTL9 are expected to provide fertile grounds for

discussing the role of statistical modeling in promoting (or hindering) students’ emergence of

aggregate view of data:

1. How do students’ articulations of aggregate view emerge while they explore data in

an attempt to model a real phenomenon?

2. What might be the relations between the development of ideas and concepts about

statistical models and modeling and the development of aggregate view of data?

3. How can reasoning about modeling and aggregate views of data in the context of ISI

be further developed in primary level?

4. What was the role of TP2 in the shaping of students’ aggregate views of data and

models?

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i This study was supported by the British Academy Small Research Grant Scheme (SG112288). The views expressed in this

paper do not necessarily reflect the views or policy of the British Academy.