Institut für vergleichende Studien
Fakultät für Maschinenbau

Developing early statistical thinking
with primary school students
and early inferential thinking
with preservice teachers forprimary school
Special lecture at JCOTS 2020
Institute of Statistical Mathematics,
TachikawaDr. Daniel Frischemeier
University of Paderborn
1Institut für vergleichende Studien
Fakultät für Maschinenbau

Overview of my lecture
1. Developing thinking about data of primary school students
using TinkerPlots
2. Developing thinking about uncertainty of primary school
students using TinkerPlots
3. Developing thinking about uncertainty of preservice
teachers for primary school (Randomization tests withTinkerPlots)

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Part 1Developing reasoning about data of primary school
students using TinkerPlots3

Institut für vergleichende Studien
Fakultät für Maschinenbau

The need for Early Statistical Thinking„Today‘s students need to learn
to work and think with data andchance from an early age, sothey begin to prepare for thedata-driven society in which theylive.“ (Ben-Zvi, 2018, vii)

Leavy, A., Meletiou-Mavrotheris, M., & Paparistodemou, E. (2018).
Statistics in Early Childhood and Primary Education: Supporting Early
Statistical and Probabilistic Thinking. Singapore: Springer.

4Institut für vergleichende Studien
Fakultät für Maschinenbau

Starting point
Proposals to promote statistical thinking at primary level• Support students in generating adequate statistical questions (u.a. Allmond & Makar,
2010; Arnold, 2013)• Exploring data with digital tools (u.a. Konold & Higgins, 2003; Harradine & Konold,
2006; Konold, 2007)• Working with meaningful data (u.a. Leavy & Hourigan, 2018)• Developing global perspectives on distributions (u.a. Bakker, 2004; Bakker &Gravemeijer, 2004; Konold et al., 2015)• Comparing groups with regard to preformal concepts like modal clumps (u.a.
Konold et al., 2002; Bakker, 2004; Fielding-Wells, 2018; Allmond & Makar, 2018)• Conducting real statistical projects, experiencing data analysis cycle (z.B. Wild
& Pfannkuch, 1999)5

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Paradigm in statistics education – also in primary school
Data analysis cycle like PPDAC-Cycle (Wild & Pfannkuch, 1999)

Different phases of PPDAC• Problem
Developing statistical question• PlanPlan of data collection• Data Collection

Data collection (survey, observation, experiment)• Analysis

Data analysis and exploration• Conclusionsof findings
Is it possibleInterpretation
to go beyond
this in primary school?
for example
Exploring large and real datasets? (u.a. Garfield & Ben-Zvi, 2008)
Using digital tools for data exploration? (u.a. Konold, 2006)
 Conducting statistical projects with real and meaningful data?

6

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Learning data analysis with digital tools

1. Building understanding for fundamental data operations (likeSeparate and Stack) on the example of small data sets
 Data analysis with Datacards (c.f. Harradine & Konold, 2006)

2. Transferring activities from 1. to the analysis of larger datasets
 Data analysis with TinkerPlots (c.f. Konold, 2006)

7

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Data cards (Biehler & Frischemeier, 2015)
Fantasy namegenderEye color

femaleblue

Brother, sisters
height
Story told before sleeping

noShoe size
8Institut für vergleichende Studien
Fakultät für Maschinenbau

Data cards without any order

Data: (Primary school Willich, class 4, 2017)9

Institut für vergleichende Studien
Fakultät für Maschinenbau

Data cards separated with regard to „how students come to school“
By car

By busBy bikeBy feetData: (Primary school Willich, class 4, 2017)10

Institut für vergleichende Studien
Fakultät für Maschinenbau„Data cards“ bar graph
By car

By busBy bikeBy feetData: (Primary school Willich, class 4, 2017)11

Institut für vergleichende Studien
Fakultät für Maschinenbau„Data cards“ bar graph (framed bars)
By car

By busBy bikeBy feetData: (Primary school Willich, class 4, 2017)12

Institut für vergleichende Studien
Fakultät für Maschinenbau

Conventional bar graph
By car

By busBy bikeBy feetData: (Primary school Willich, class 4, 2017)13

Institut für vergleichende Studien
Fakultät für Maschinenbau

From data cards to TinkerPlots
 To build on the data analysis with data cards and on
well-known data operations (separate, stack) Enabling young students to explore large datasets

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Institut für vergleichende Studien
Fakultät für Maschinenbau

TinkerPlots• Developed by Clifford Konold and Craig Miller on the basis of current
findings in statistics education didactics for use in classes 3-8• TinkerPlots is suitable to provide early access to statistical andprobabilistic ways of thinking of students (Konold, 2007) - the special
issue about it:• Takes up the work with data cards• Creating the graphics using three operations: "Separate“, “Stack”
and “Order”
Demo

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Institut für vergleichende Studien
Fakultät für Maschinenbau

TinkerPlots
At which stages can TinkerPlots support young learners in their data
analysis activities?
• Transferring the data operations (Separate, Stack, etc.) to larger
datasets and creating conventional diagrams in larger datasets (Konold
2006)• Creating pie charts• Paving the way to stacked dot plots (u.a. Cobb 1999, Bakker 2004) 
distribution of a numerical variable• Using preconcepts like modal clumps (Konold et al. 2002, Bakker
2004) and hatplots (Watson et al. 2008), to read and interpret
distributions of numerical variables
16Institut für vergleichende Studien
Fakultät für Maschinenbau

Stacked dot plots
How can we visualize the distribution of a numericalvariable (like height) in primary school?

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Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

From case value plot to a stacked dot plot in four steps
Step 2: Value bars
orderedStep 1: Value bars
unordered
See Cobb (1999), Bakker (2004)18

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

From case value plot to a stacked dot plot in four steps

Step 4: Stack dots

Step 3: Replacing value
bars with dots
19

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Local vs. global view on distributions (see Bakker & Gravemeijer,
2004)

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Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Local vs. global view on distributions(see Bakker & Gravemeijer, 2004)

Distribution (global view)

Data (local view)(see Bakker & Gravemeijer, 2004)21

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Modal clumps“Main interval” in the data

Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A., Wing,
R., & Mayr, S. (2002). Students' use of modal clumps to summarize
data. Paper presented at the Sixth International Conference on
Teaching Statistics, Cape Town, South Africa.

• “range of data in the heartof a distribution of values”• “appear to allow students
to express simultaneously
what is average and how
variable the data are”
• “may provide useful
beginning points for
explorations of more formal
statistical ideas of center“(Konold et al. 2002, p. 1).
22

Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

A hat for the data
Hatplot visualizes the middle 50% of the data
• can offer an easier approach toidentifying spread in
distributions since they arecomposed of three major parts(two brims and a crown):• the brim is a line that extends to
the range for each group; thecrown is a rectangle that, […]shows the location of the middle
50% of the data – theInterquartile Range (IQR)(Konold 2002, p. 1).
Watson, J., Fitzallen, N., Wilson, K., & Creed, J. (2008). The
Representational Value of HATS. Mathematics Teaching in Middle
School, 14(1), 4-10.

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Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Methaphorical hats: Describing and comparing distributions

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Design and realization of a teaching unit to enhance
statistical reasoning (especially with regard to groupcomparisons) in primary school25

Institut für vergleichende Studien
Fakultät für Maschinenbau

Outline of our research project
Goals• introduce primary school students to modal clump and hatplots
and lead them towards a global view on distribution• to provide them with a first concept for comparing groups.
 Design and the realization of lessons leading to groupcomparison activities in grade 4 (age: 10-11 years) in primaryschool.
 Accompanying study which assesses the performance ofstudents before and after attending to a teaching unit aboutcomparing groups.
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Institut für vergleichende Studien
Fakultät für Maschinenbau

Research Methodology
Didactical Design Research (Prediger, & Zwetzschler, 2013).

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Design of the teaching unit
The lessons were designed and taught by a preservice teacher for
primary school within the scope of her Bachelor Thesis
Major design ideas:• Elements of the „Statistical Reasoning Learning Environment“: Cobb &McClain (2004) and Garfield & Ben-Zvi (2008)• PPDAC-cycle (Wild & Pfannkuch 1999)• Working with real data (Engel 2007, Garfield & Ben-Zvi 2008)• Using educational software TinkerPlots (Konold 2007, Garfield & Ben-Zvi 2008,
Ben-Zvi & Pfannkuch 2011)28

Institut für vergleichende Studien
Fakultät für Maschinenbau

Realization of the course (Cycle 2)• Primary school in rural area in Germany, 4th gradestudents (ages 10-11)• 10 sessions à 45 minutes• Participants: 12 grade 4 students (ages 10-11)29Institut für vergleichende Studien
Fakultät für Maschinenbau

Overview of the lessons of the teaching unit

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Applying PPDAC cycle in the teaching learning environment
Conclusions, Data posters

Posing statistical questions

Data analysis withTinkerPlots,
Comparing groups withTinkerPlots

Creating a questionaire for grade 3
students, collecting data in class

Data cleaning, Import of thedata in TinkerPlotsInstitut für vergleichende Studien
Fakultät für Maschinenbau

Contents of the lessons on comparing groups

1. Learning stepwise to compare groups• Reading the data and reading between the data on stacked dot
plots, Pupils get to know modal clumps to identify characteristicsof the distribution• Pupils learn to use modal clumps, medians and hatplots tocompare groups

2. Pupils working on group comparison tasks

32Institut für vergleichende Studien
Fakultät für Maschinenbau

Learning to compare groups
First: Learning to reason about one distribution of a numerical variable

Distribution of height in small dataset on
board with magnetic dots

Distribution of height in small dataset in
TinkerPlotsDistribution of height inlarger dataset withmodal clump in
TinkerPlots
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Institut für vergleichende Studien
DanielProf.
Dr. Hans-Michael& RolfDietz
Biehler
Fakultät
fürFrischemeier
Maschinenbau

Comparing distributions

Identifying modal clumps

Identifying median in modal clumps

Using hatplots
34

Institut für vergleichende Studien
Fakultät für Maschinenbau

Contents of the lessons on comparing groups

1. Learning stepwise to compare groups• Reading the data and reading between the data on stacked dot
plots, Pupils get to know modal clumps to identify characteristicsof the distribution• Pupils learn to use modal clumps, medians and hatplots tocompare groups
2. Pupils working on group comparison tasks• Pupils generate statistical questions leading to groupcomparisons and conduct group comparisons with given data
• Pupils conduct group comparisons with given data anddocument their findings on posters.
• Pupils present their findings to their classmates
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Institut für vergleichende Studien
Fakultät für Maschinenbau

Impressions from the teaching unit

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Exploration of data with TinkerPlots in pairs

37Institut für vergleichende Studien
Fakultät für Maschinenbau

Presentation of data posters

38Institut für vergleichende Studien
Fakultät für Maschinenbau

Data posters

39Institut für vergleichende Studien
Fakultät für Maschinenbau

Presenting all findings in school40

Institut für vergleichende Studien
Fakultät für Maschinenbau

Design-based Research: Collected data
• Notebooks of students• TinkerPlots files of students• Data posters• Pre- / post test
• Feedback survey• Interviews with selected students

41

Institut für vergleichende Studien
Fakultät für Maschinenbau

Design-based Research: Collected data
• Notebooks of students• TinkerPlots files of students• Data posters• Pre- / post test
• Feedback survey• Interviews with selected students

42

Institut für vergleichende Studien
Fakultät für Maschinenbau

Research Questions

In which way does statistical reasoning with regard to groupcomparisons of the students improve after the course?
In which way does the course have an impact on the attitudes of theparticipants?

43

Institut für vergleichende Studien
Fakultät für Maschinenbau

Design of study

Examine group comparison strategies of primary school
students before and after they were taught about groupcomparisons in our teaching unit
For the evaluation we gave the students a group comparison task
before they have attended our teaching unit and we gave themthe same task after they have attended our teaching unit two
weeks later.

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Institut für vergleichende Studien
Fakultät für Maschinenbau

The task
Do first graders or third graders tend to have heavier backpacks?
Explain your answer.

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Participants, Data collection and Data analysis

Participants• 12 grade four students (age 10-11)• All of them have participated in the teaching unit
Data collection• written notes on the task from all students (n=12 before; n=12
after)
Method for data analysis• Structured qualitative content analysis (Mayring 2015) andmixed approach of coding (Kuckartz 2012) to identify thegroup comparison elements used46

Institut für vergleichende Studien
Fakultät für Maschinenbau

Deriving group comparison items: coding procedure• Mixed approach (Kuckartz 2012)  we generated the categories(group comparison items used) in a first step from a deductive
point of view and then – in a second step – refined them
inductively.

• Deductive point of view (first step)  Biehler (2007),
Pfannkuch (2007), Frischemeier (2017): Center, Spread,
Skewness, Shift, p-based and q-based.
• No q-based comparisons because (too sophisticated forprimary school pupils).
• With regard to comparisons of center, we also added “total
score” of two groups as comparison element, which is asustainable element when the two groups are equal-sized. 47

Institut für vergleichende Studien
Fakultät für Maschinenbau

Deriving group comparison items: coding procedure• Use of preconcepts like modal clumps (deductive)• Use of precursor visualisations like hatplots (deductive)• Use of shift of points (inductive)• Another possibility could be to identify the shift of the points to
the right in the group of third graders in comparison to thegroup of first graders (exemplary coding: “Because the points
are located further behind”, see Maria, Figure 7). We call thiskind of comparison strategy “shift of points”.

48Institut für vergleichende Studien
Fakultät für Maschinenbau

Results

8/12

4/12

12/12

11/12
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Institut für vergleichende Studien
Fakultät für Maschinenbau

Student answers: The case of Maria,
Pretest (shift of points):

Posttest (median and hats):

50

Institut für vergleichende Studien
Fakultät für Maschinenbau

Design-based Research: Collected data
• Notebooks of students• TinkerPlots files of students• Data posters• Pre- / post test
• Feedback survey• Interviews with selected students

51

Institut für vergleichende Studien
Fakultät für Maschinenbau

Research Questions

In which way does statistical reasoning with regard to groupcomparisons of the students improve after the course?
In which way does the course have an impact on the attitudes of theparticipants?

52

Institut für vergleichende Studien
Fakultät für Maschinenbau

Survey: Students attitudes towards specific components ofthe teaching unit

N=12 students filled out the survey after the teaching unit

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Summary of results• The study offers interesting insights in fourth graders´ groupcomparison strategies for the further development of groupcomparison activities.  it is possible to establish first notations ofcomparing groups in primary school classroom (after introducing
basics of data analysis, stacked dot plots, etc.).
• Even in the pretest, most of the students have shown that they wereable to decide whether third graders have heavier backpacks
compared to first graders, but the group comparison items of many of
the students have been inadequate or have been focusing on local
features.
• After attending to the teaching unit the quality of group comparison
elements used has improved massively.
• All students show a positive attitude towards statistics and theteaching unit on comparing groups with TinkerPlots
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Institut für vergleichende Studien
Fakultät für Maschinenbau

Some Implications…
• TinkerPlots is an adequate digital tool to introduce youngstundents in the analysis of larger datasets and into moresophisticated statistical activities (like e.g. group comparisons)• Use real and meaningful datasets can and should already be
implemented in primary school classroom• Connect data analysis with TinkerPlots with enactive data
analysis with data cards, so that both processes “learning data
analysis” and “learning the software for analysing data” areconnected• To develop a global view on distributions and to facilitate thegroup comparison process, the stepwise use of modal clumps,
medians and hats is very promising
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Institut für vergleichende Studien
Fakultät für Maschinenbau

Part 2Developing reasoning about uncertainty of primaryschool students using TinkerPlots

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Institut für vergleichende Studien
Fakultät für Maschinenbau

The need for Early Statistical Thinking„Today‘s students need to learn
to work and think with data andchance from an early age, sothey begin to prepare for thedata-driven society in which theylive.“ (Ben-Zvi, 2018, vii)

Leavy, A., Meletiou-Mavrotheris, M., & Paparistodemou, E. (2018).
Statistics in Early Childhood and Primary Education: Supporting Early
Statistical and Probabilistic Thinking. Singapore: Springer.

57

Institut für vergleichende Studien
Fakultät für Maschinenbau

A conceptualization of randomness

Batanero (2015) distinguishes between three
conceptualizations of randomness:(i) Randomness as equiprobability,(ii) randomness as stability of frequencies,(iii) subjective view of randomness.

58

Institut für vergleichende Studien
Fakultät für Maschinenbau

Conceptualizations of randomness• In “randomness as equiprobability” probability is defined in the sense of
Laplace as “the number of favorable cases to a particular event divided
by the number of all cases possible in that experiment, provided all thepossible cases are equiprobable” (Batanero, 2015, p. 36).
• “Randomness as stability of frequencies” is related to the empirical law oflarge numbers, probability is defined according to Batanero (2015, p. 37)
as “the hypothetical number towards which the relative frequency tends”.
• The subjective view “considered probability as a personal degree of belief
that depends on a person´s knowledge or experience” (Batanero, 2015,
p. 37).
59

Institut für vergleichende Studien
Fakultät für Maschinenbau

Conceptualizations of randomness• In “randomness as equiprobability” probability is defined in the sense of
Laplace as “the number of favorable cases to a particular event divided
by the number of all cases possible in that experiment, provided all thepossible cases are equiprobable” (Batanero, 2015, p. 36).
• “Randomness as stability of frequencies” is related to the empirical law oflarge numbers, probability is defined according to Batanero (2015, p. 37)
as “the hypothetical number towards which the relative frequency tends”.
• The subjective view “considered probability as a personal degree of belief
that depends on a person´s knowledge or experience” (Batanero, 2015,
p. 37).
60

Institut für vergleichende Studien
Fakultät für Maschinenbau

Conceptualizations of randomness• In “randomness as equiprobability” probability is defined in the sense of
Laplace as “the number of favorable cases to a particular event divided
by the number of all cases possible in that experiment, provided all thepossible cases are equiprobable” (Batanero, 2015, p. 36).
• “Randomness as stability of frequencies” is related to the empirical law oflarge numbers, probability is defined according to Batanero (2015, p. 37)
as “the hypothetical number towards which the relative frequency tends”.
• The subjective view “considered probability as a personal degree of belief
that depends on a person´s knowledge or experience” (Batanero, 2015,
p. 37).
61

Institut für vergleichende Studien
Fakultät für Maschinenbau

Goals of this project

Design of a teaching unit which covers each of the three
conceptualizations of randomness to develop primaryschool students´ reasoning about uncertainty

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Research Methodology
Didactical Design Research (Prediger, & Zwetzschler, 2013).

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Institut für vergleichende Studien
Fakultät für Maschinenbau

Research study• First cycle of Design research• Primary school in rural area in Germany• 7 lessons (45 minutes each)– Data: field notes of the teacher, documents of students,
videos• Participants: 20 students, grade 4 (10-11 years old)
– no specific pre-knowledge in statistics apart fromcollecting data in tallies and creating and reading bar
charts and reading pie charts.

Institut für vergleichende Studien
Fakultät für Maschinenbau

Design ideas for the teaching learning environment• Taking into account all three conceptualizations of randomnessof Batanero (2015) to develop the reasoning about uncertainty
among grade 4 students.
• Elements of the „Statistical Reasoning Learning Environment“(Garfield & Ben-Zvi , 2008), e.g.
• Peer learning settings, promoting classroom discourse• Using educational software TinkerPlots (Konold 2007, Garfield& Ben-Zvi 2008, Ben-Zvi & Pfannkuch 2011)  TP Sampler

65Institut für vergleichende Studien
Fakultät für Maschinenbau

Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
66

6-7

Institut für vergleichende Studien
Fakultät für Maschinenbau

Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
67

6-7

Institut für vergleichende Studien
Fakultät für Maschinenbau

Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
68

6-7

Institut für vergleichende Studien
Fakultät für Maschinenbau

Contents of the second lesson• Students evaluate events of several chance experiments using
the expressions “certain”, “probable”, “unlikely” and “impossible”.
• Teacher offers a strip with a scale from “impossible” to “certain”
which the students use to show and to visualize their expectation
with regard to specific events.
• Then: Teacher expands the scale and adds frequency
descriptions like “never”, “seldom”, “often” and “always” for eachterminology

69

Institut für vergleichende Studien
Fakultät für Maschinenbau

Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
70

6-7Institut für vergleichende Studien
Fakultät für Maschinenbau

Contents of the third lesson
Pupils conduct hands-on experiment “throw of a dice” and collectdata to get the insight that the probability for each side of the dice isequal
Students work in pairs (10 pairs) to throw a dice fifty times anddocument the frequencies of the occurrences of each side (handson activities) Teacher uses notations like “In 79 of 500 cases the side 1
has occurred” or “in 78 of 500 cases the side 2 has occurred”
to make the frequencies comparable
 Insight for the students: the probability for each side of thedice is equal.

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Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
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Contents of the fourth lesson

Students are introduced in the simulation of chance experimentswith the TinkerPlots sampler.
 Teacher demonstrates how to realize the “toss of a coin” in the
TinkerPlots sampler

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Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
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Contents of the fifth lesson• Teacher chooses a more complex chance experiment (“throw of
two dice”)  multi-step chance experiments.
• Task for the students: “Find out which sum is more likely toappear when throwing two dice.”• Students are paired in groups and each group was given twodice and the task to throw the two dice fifty times and to
document their outcomes  teacher collects the results of allten groups
The first insight for the
students in lesson 5 isthat the different sumsof the throw of twodice are notequiprobable75

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Seven lessons of the teaching learning environmentNo

1
2Content of lesson

Pupils learn about how to read and interpret statistical bar graphs and pie
chartsPupils estimate and compare probabilities of events by classifying them in“certain”, “possible”, “unlikely” and “impossible”.

3

Pupils conduct hands-on experiment “throw of a dice” and collect data to get
the insight that the probability for each side of the dice is equal.

4

Pupils are introduced in the sampler of TinkerPlots and get first experiencesof the empirical law of large numbers by simulating the throw of a coin with the
TinkerPlots sampler

5

Pupils conduct hands-on experiment “throw of two dice”, collect the sum of
the two dice for each throw and collect data to use the frequencies of occurrenceof the different sums to make statements with regard to the probability of theevents of different sums
Pupils conduct TinkerPlots simulation “throw of two dice” and collect datato compare the probability of events. Pupils try to find explanations why somesums appear more frequent than other sums when throwing two dice.
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Contents of the sixth and seventh lesson• The teacher reflects the results from the hands-on activity in thefifth lesson on the chance experiment “sum of two dice” anddemonstrates how to set up this chance experiment in the
TinkerPlots sampler

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Contents of the sixth and seventh lesson• The teacher repeats the simulation in TinkerPlots several times
and the students observe that the mode of the distribution is thesum “7”.
• The students have the idea to consider the possible outcomes of
the throw of two dice for each of the sums (2-12)  identify theoutcomes for each sum and collect their results• The students explore that the sum “7” has the most outcomes(1+6, 6+1, 2+5, 5+2, 3+4, 4+3) and therefore occurs more often
in the experiment than for example the sum “3” (with the twooutcomes 1+2 and 2+1) or the sum “2” (with the outcome 1+1).
• The students also find out the outcomes for each sum and that
there are 36 outcomes in total (each of them are equiprobable).
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Contents of the sixth and seventh lesson• Comparisons of probabilities of certain events like “the sum 6 is more
likely to show up than the sum 3”
 because there are five favorable out of 36 outcomes for sum “6”(1+5, 5+1, 2+4, 4+2, 3+3) but only two favorable out of 36 outcomesfor sum “3” (1+2, 2+1).

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First observations from the teaching-learning environment• Students were very engaged - especially with the simulation ofchance experiments (hands-on, TinkerPlots)• Elements of the three conceptualizations of randomness (seeBatanero, 2015) can be already implemented in primary schoolclassroom.
• TinkerPlots sampler seems to be a powerful educational tool tofacilitate the modelling and the data production process whensimulating chance experiments in primary school
 Some students needed support (especially when setting up themodel)
 Challenge: interpretation of the produced data

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Explorative Study: „Primary school students‘
statistical reasoning when conducting chance
experiments with TinkerPlots“81Institut für vergleichende Studien
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Research question

In which way does the teaching unit develop the competence of pupils
with regard to compare probabilities of events of multi-step chance
experiments?

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Data, participants and data analysis
We collected data on different levels:(a) written pre/post-tests,(b) working notes on tasks and activities after each lesson and(c) interviews after the teaching unit.

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Data, participants and data analysis

We collected data on different levels:(a) written pre/post-tests,(b) working notes on tasks and activities after each lesson and(c) interviews after the teaching unit.
Written pre/post tests were handed out at the beginning and at the end of
the teaching unit Six tasks.
 Posttest is identical to the pretest.
 19 students have participated in the pre- and posttest.
 We will only concentrate on the tasks which focus on the comparisonof probabilities of events of multi-step chance experiments (tasks 5
and 6).
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Tasks
Task (5) You and your friend Tim play a game where you toss a coin
twice. You win if you get tail both times. Tim wins if he gets tail and head
when tossing the coin twice. Do you both have the same chances of
winning? Explain!
Task (6) Anna and Tom are playing a game. They throw a dice twice. If
the sum of the two dice is "7", Anna wins. If the sum is 3 or 12, Tom
wins. Do Anna and Tom have the same chance of winning? Explain!

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Data analysis procedure & Results
1. Coding the correctness of the statements of each participant in pretest and post-test
2. Frequency analysis and calculation of the percentages of correctstatements of each task in comparison to pre- and posttest.
Item% pretestn% posttestnDifference
n

5

0.0%
0

47.4%
9

+47.4
+9

6

9.5%
2

52.6%
10

+43.1
+8
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Exemplary statements of students (task 5)
Task (5): You and your friend Tim play a game where you toss a coin twice. You
win if you get tail both times. Tim wins if he gets tail and head when tossing thecoin twice. Do you both have the same chances of winning? Explain!

“Yes, we have an equal chance of
winning.” (statement in pre-test)“Tim has a better chance
to win because I have
one chance to win andTim has two.” (statementin post-test)87

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Exemplary statements of students (task 6)
Task (6): Anna and Tom are playing a game. They throw a dice twice. If thesum of the two dice is "7", Anna wins. If the sum is 3 or 12, Tom wins. Do Anna
and Tom have the same chance of winning? Explain!

“Tom will win because he has two
numbers and has a better chance
to win.” (statement in pre-test)“Anna will win more
often than Tom, because
the 7 has more
possibilities with twodice than the 3 and 12.”(statement in post-test) 88

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Summary• Elements of the three conceptualizations of randomness(see Batanero, 2015) can be already implemented inprimary school classroom.
• The performance of the students from pre-test to post-test
has increased considerably• The selected statements in the post-test show that thestudents´ reasoning about the interpretation of events in
multi-step chance experiments has developed in a positiveway• Some students are now able to use, e.g. elements of
the approach of Laplace to explain their statements
when comparing the probability of events.
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First „Implications“ (although not in a sense of experimental evidence)• Statistical reasoning should be initiated as early as
possible  in primary school• The TinkerPlots sampler seems to be a powerfuleducational tool to facilitate the modelling and the data
production process when simulating chance experiments
in primary school  challenge: interpretation of the
produced data (decision making based on data)• The students should conduct the chance experiments as
hand-on activities first and then use software like
TinkerPlots for the modelling and data production
process.
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Outlook• The teaching unit cannot only be of interest for primaryschool researchers and teachers but can also be
adapted for implementation in secondary schoolclassroom.
• The other data (working notes and interview data)collected in the frame of this research project are still
under analysis.
• Interview: insight into the cognitive processes ofprimary school students when modelling one/multistep chance experiment with TinkerPlots91

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Part 3Developing reasoning about uncertainty ofpreservice teachers for primary school(Randomization tests with TinkerPlots)

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Short review of relevant literature

Reasoning in uncertainty• Problems occur when a result found in a sample has to begeneralized –> Randomization tests as an opportunity to makegeneralizations (Rossman, 2008)• Cobb (2007): the 3 „Rs“: Randomize data production, Repeat bysimulation and Reject any model that puts your data in its tail93

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Use of randomization tests in German school curriculum• Randomization tests in textbooks are widely unknown atsecondary and tertiary level• Even hypothesis testing with p-values appears seldomly• Formal hypothesis testing with pre-defined significancelevels is a topic in most federal states, but often not
obligatory
Research on how German preservice teachers cope withrandomization tests see Frischemeier & Biehler (2014);
Biehler, Frischemeier & Podworny (2015) and Podworny(2019)

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Schematic model for doing a randomization test in a real worldcontext

See Biehler, Frischemeier &Podworny (2015)95

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Randomization tests with TinkerPlots
Why TinkerPlots?
• Easy introduction to chance experiments via the sampler• Powerful tool for modelling• No formulas need
• Making a statistical process visible (drawing out of boxes;

random assignment; no black box) (Schnotz, 2002)• TinkerPlots as a „data factory“ for producing a new
sample  Understanding distributions by modeling them

(Konold et al., 2008)96

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An example:
Randomization test with TinkerPlots
Context:
Caffeine vs. non Caffeine (clinical study)
Does caffeine facilitate rapid movement?

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Research question:
Does caffeine produces an increase in the average tap
rate?

Example from Lock etal. (2013)98

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Difference of means of the two groups in the real data

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Formulating an adequate null hypothesis
The group membership caffeine vs. non-caffeine has noeffect on the fingertip rate

The difference between caffeine and non-caffeine users has
occured at random Idea: We simulate this Hypothesis in TinkerPlots
We investigate, how likely is it to get a difference of means
3.5 or larger in the case of randomly allocated groups(caffeine vs. non-caffeine)100Institut für vergleichende Studien
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Setting up the model in TinkerPlots
RepeatReal data„Label“Draw

without replacement101

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Running the TinkerPlots sampler

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Comparing two „random“ distributions after the labelling process103

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Collecting measures
Collect

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Identifying and interpreting p-value

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Identifying and interpreting p-value

This shows clear evidence against our hypothesis that "the group membership caffeine vs. nocaffeine has no effect on the fingertip rate".
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Statistical steps of the randomization test - Overview

Step 1: Reading off the difference of the means of the groups
in the dataset
Step 2: Formulating an adequate null hypothesis
Step 3: Describing the null model
Step 4: Formulating the test statistic
Step 5: Determining the p-value
Step 6: Drawing conclusions from the p-value

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Typical problems & Implications
Problems cf. Frischemeier & Biehler (2014); Frischemeier (2017):• Generation of adequate Null hypothesis• Identifying p-value correctly• Interpreting p-value correctly
Implications cf. Frischemeier & Biehler (2014); Frischemeier (2017):• Discussing different null hypotheses (good vs. bad examples)• Realizing hands-on allocation process before technical introduction
with TinkerPlots• Providing a scheme with the several steps of a randomization test
• Providing a guide how to interpret p-values

 TinkerPlots as powerful tool to realize randomization tests

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Supporting material: Randomization test scheme

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Supporting material: Handout „evidence“ - language

Hand-out:*We have weak evidence against the null hypothesis, if p ≤ 10%.
*We have medium evidence against the null hypothesis, if p ≤ 5%.
*We have strong evidence against the null hypothesis, if p ≤ 1%.
*We have very strong evidence against the null hypothesis, if p ≤ 0.1%.

*We have no evidence against the null hypothesis, if p > 10 %.

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Final thoughts on my lecture111

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Thank you very much for your attention!
dafr@math.upb.de112

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Literature (Part 1, chosen)
•
•
•
•
•
•
•
•
•
•
•
•

Bakker, A. (2004). Design research in statistics education - on symbolizing and computer tools. (Dissertation), University of
Utrecht.
Bakker, A., & Gravemeijer, K. (2004). Learning to reason about distributions. In D. Ben-Zvi & J. Garfield (Eds.), The challenge ofdeveloping statistical literacy, reasoning and thinking (pp. 147–168). Dordrecht: Kluwer Academic Publishers.
Burrill, G., & Biehler, R. (2011). Fundamental statistical ideas in the school curriculum and in training teachers. In C. Batanero,
G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics-Challenges for teaching and teacher education (pp.
57–69). Dordrecht/Heidelberg/London/New York.
Konold, C. (2002). Hat Plots?. Unpublished manuscript. University of Massachusetts.
Konold, C. (2007). Designing a data tool for learners. In M. Lovett & P. Shah (Eds.), Thinking with data: The 33rd Annual
Carnegie Symposium on Cognition (pp. 267–292). Hillside: Lawrence Erlbaum Associates.
Konold, C., & Higgins, T. L. (2003). Reasoning about data. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A ResearchCompanion to Principles and Standards for School Mathematics (pp. 193–215). Reston: National Council of Teachers ofMathematics.
Konold, C., & Miller, C. (2011). TinkerPlots 2.0. Emeryville: Key Curriculum Press.
Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in
Mathematics Education, 33(4), 259–289.
Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A. D., Wing, R., et al. (2002). Students’ use of modal clumps tosummarize data. In B. Phillips (Ed.), Proceedings of the International Conference onTeaching Statistics [CD-ROM], Cape Town.
Voorburg: International Statistics Institute.
Leavy, A.,Meletiou-Mavrotheris,M., & Paparistodemou, E. (2018). Statistics in early childhood and primary education:
Supporting early statistical and probabilistic thinking. Singapore: Springer.
Watson, J., Fitzallen, N., Wilson, K., & Creed, J. (2008). The representational value of HATS. Mathematics Teaching in Middle
School, 14(1), 4–10.
Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223–248.

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Literature (Part 2, chosen)• Batanero, C. (2015). Understanding Randomness: Challenges for Research and Teaching. In K. Krainer& N. Vondrova (Eds.), Proceedings of the 9th Congress of the European Society for Research in
Mathematics Education (pp. 34–49). Charles University in Prague: Faculty of Education and ERME.
• Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design Experiments in Educational
Research. Educational Researcher, 32(1), 9-13.
• Garfield, J., & Ben-Zvi, D. (2008). Developing Students’ Statistical Reasoning. Connecting Research andTeaching Practice. The Netherlands: Springer.
• Hasemann, K., & Mirwald, E. (2012). Daten, Häufigkeit Und Wahrscheinlichkeit. In G. Walther, M. vanden Heuvel-Panhuizen, D. Granzer, & O. Köller (Eds.), Bildungsstandards Für Die Grundschule:
Mathematik Konkret (pp. 141-161). Berlin: Cornelsen Scriptor.
• Jackes-Schulte, A. (2016). Design, Durchführung Und Evaluation Einer Unterrichtsreihe Zur Entwicklung
Der Kompetenz "Gewinnchancen Beim Doppelten Würfelwurf Einzuschätzen" in Jahrgangsstufe 4.
(Bachelor of Education), University of Paderborn.
• Konold, C., & Miller, C. (2011). Tinkerplots 2.0. Emeryville, CA: Key Curriculum Press.
• Leavy, A., Meletiou-Mavrotheris, M., & Paparistodemou, E. (2018). Statistics in Early Childhood andPrimary Education: Supporting Early Statistical and Probabilistic Thinking. Singapore: Springer.
• Plückebaum, K. (2018). Simulieren Von Zufallsexperimenten in Der Primarstufe Mit Der Software
Tinkerplots. (Master of Education), Universität Paderborn, Paderborn.
• Weustenfeld, W. (2007). Die Augensumme Zweier Würfel Voraussagen: Alles Nur Eine Frage Von Glück
Oder Pech. Stochastik in der Schule, 27(3), 2-15.
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Literature (Part 3, chosen)• Biehler, R.; Frischemeier, D. & Podworny, S. (2015). Preservice Teachers´ Reasoning about
Uncertainty in the Context of Randomization Tests. In A. Zieffler & E. Fry (Eds.): Reasoning about
Uncertainty: Learning and Teaching Informal Inferential Reasoning (S. 129-162). Catalyst Press..
• Cobb, G. (2007). The introductory statistics course: A Ptolemaic curriculum? Technology
Innovations in Statistics Education, 1(1).
• Frischemeier, D. & Biehler, R. (2014). Design and exploratory evaluation of a learning trajectory
leading to do randomization tests facilitated by TinkerPlots. In: B. Ubuz, C. Haser & M. A. Mariotti(Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics
Education (pp. 799-809).
• Frischemeier, D. (2017). Statistisch denken und forschen lernen mit der Software TinkerPlots.
Wiesbaden: Springer Spektrum.
• Hand, D. J., Daly, F., McConway, K., Lunn, D., & Ostrowski, E. (1993). A handbook of small data
sets. Chapman and Hall: London.
• Konold, C. & Miller, C. (2011). TinkerPlots TM Version 2 [computer software]. Emeryville, CA: Key
Curriculum Press.
• Lock, R. H., Lock, P. F., Lock, K., Lock, E. & Lock, D. (2013). Statistics: Unlocking the power ofdata. Wiley Global Education.
• Rossman, A. (2008). "Reasoning about Informal Statistical Inference: A Statistician´s View."
Statistics Education Research Journal. 7(2): 5-19.
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