CARMEN BATANERO

Univers idad de Granada , España

www.ugr.es /~ba tanero /

UNDERSTANDING

RANDOMNESS Challenges for Research and Teaching

MOTIVATION

Pervasiveness of randomness (Hacking, 1990)

A basic concept, yet

implicit in teaching

and subjected to

controversies

Variety of

misconceptions and

wrong intuitions

Probability in

school curricula

European contribution to research

2

SCHEME

1. From chance to randomness

2. Formalizing randomness

3. Personal views of randomness

4. Teaching and learning the idea of randomness

5. Final reflections

3

1. FROM CHANCE TO RANDOMNESS

The idea of chance is as old as civilization.

Chance mechanisms were used to predict the future and to engage in decision-making in many civilizations.

A scientific idea of randomness was absent until the beginning of the Middle Ages (Bennet, 1993).

Different conceptions of chance have been used to describe uncertain situations (Fine, 1971; Bennet, 1993; Batanero

et al, 1998, 2005; Borovcnik, & Kapadia, 2014; Lahanier-Reuter, 1998).

Some of them still appear in students

and teachers

(Engel & Sedlmeier, 2002; Batanero et al., 2014).

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Randomness - Causality

Intuitive meaning of randomness (Moliner, 2000)

Random: “Uncertain. It is said of what depends on

luck or chance”.

Chance: “Presumed cause of events that are neither

explained by natural necessity nor by a human or

divine intervention”.

In a first historical phase 'random‘ was opposed to that

whose causes were known, and 'chance' was personified

as the cause of random phenomena (Bennet, 1993).

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Deterministic view of the world

A slight variation in this meaning is believing that every

phenomenon has a cause.

“Everything is the combined fruit of chance and

need” (Democritus).

“Nothing happens at random; everything happens out

of a reason and by necessity” (Leucippus).

Aristotle considered that chance results from the

unexpected coincidence of a series of independent

events, so that the eventual result is pure chance. 6

Randomness - Causality

Deterministic thinking in the Renaissance

“It is written up there” (Diderot, 1796/1983).

“All which benefits under the sun from past, present or

future, being or becoming, enjoys itself an objective and total

certainty… since if all what is future would not arrive with

certainty, we cannot see how the supreme Creator could

preserve the whole glory of his omniscience and

omnipotence” (Bernoulli, 1713/1987, p. 14).

This conception is still found in the 19th century:

“Present events are connected with preceding ones by a link

based upon the evident principle that a thing cannot occur

without a cause which produces it” (Laplace, 1814/1995, p. vi).

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Randomness - Causality

Randomness and causality (Poincaré,1912/ 1987):

Deterministic phenomena with unknown laws.

Other phenomena, such as Brownian motion can be described by known deterministic laws, even when they are primarily random at the microscopic level.

Sometimes “a very small cause, which escapes us, determines a considerable effect that we cannot fail to see, and then we say that this effect is due to chance” (Poincaré, 1912/1987, p. 4).

Accepting fundamental chance:

Heisenberg’s uncertainty principle in quantum mechanics.

Chance is also explained by mathematical theories such as those of complexity or chaos (Morin, 1984; Ruelle, 1991).

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Modern Perception of Chance

2. FORMALIZING RANDOMNESS

Randomness formalized the idea of chance.

It was related to probability, that revealed a multifaceted character since its emergence (Hacking,

1975).

A statistical side was concerned with finding the objective mathematical rules behind sequences of random outcomes.

An epistemic side views probability as a personal degree of belief.

Some conceptions used in teaching are the classical, frequentist, subjective and axiomatic views.

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Classical Conception

Early progress in probability was linked to games of chance;

randomness was conceived as equiprobability.

An object (or event) is a random member of a given class if

there is the same probability for any other member of its class

(e.g. Cardano, 1663).

Probability is a fraction of the number of favourable cases to a

particular event divided by the number of all cases possible,

provided all the possible cases are equiprobable (de Moivre,

1718; Laplace,1814).

This definition imposes severe restrictions to the idea of

randomness (Kyburg, 1974).

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Frequentist Conception

Observed convergence in natural phenomena lead to the first developments of LLN by J. Bernoulli.

An object is a random member of a class if we can select it through a method providing a stable relative frequency in the long run.

Probability is the limit of relative frequency (von Mises,

1928; Rényi, 1966).

We never get the exact value of probability.

Sometimes it is not possible to repeat the experiment under exactly the same conditions.

The number of experiments needed is undefined.

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Subjective Conception

Bayes’ theorem transforms a prior distribution about an unknown probability into a posterior distribution.

Randomness is composed of four elements (Kyburg, 1974):

The object that is supposed to be a random member of a class;

The set where the object is a random member (population or collective);

The property for which the object is a random member of the class;

The knowledge of the person giving the judgment of randomness.

Probability is a personal degree of belief (Keynes, 1921; Ramsey, 1931; de Finetti, 1937).

The subjective character was criticized, even if the impact of the prior diminishes by objective data and de Finetti (1934/1974)

axioms.

12

Random Processes and Sequences

Throughout the 20th century, different mathematicians

formalized the mathematical theory of probability via

Kolmogorov’s axioms (1933).

The development of statistical inference and the

interest in providing algorithm to produce “pseudo-

random” sequences lead to distinguish two components

in randomness:

The generation process (random experiment);

The pattern of the random sequence produced by

the experiment (Zabell; 1992).

These components can be separated. 13

Different approaches served to define random

sequences (Fine,1971; Chaitin, 1975).

Von Mises: In an infinitely long series of outcomes, we can

find no algorithm to select a subsequence where the

relative frequency of one event is changed.

Kolmogorov: The minimal number of signs necessary to

codify a sequence provides a scale to measure its

complexity. In this approach, a sequence is random if any

coded description of the same is as long as the sequence

itself.

HTHTHTHT 4HT

HTHTTHHT 14

Random Sequences

Synthesis: Epistemic Meanings of Randomness

In the previous two approaches perfect randomness is

only a theoretical concept.

Like in probability we find different views of the

concept, that still coexist.

Using some ideas from the onto-semiotic approach

(Godino; 2002; Godino et al., 2007 ) we can describe the

differences between these epistemic meanings of

randomness.

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Onto-semiotic Configurations (Godino, 2002; 2014;

Godino, Batanero, & Font, 2007)

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Example: Classical Meaning

PROBLEMS: establishing the fair betting.

LANGUAGE: fairness, odds, game of chance, verbal and

simple algebraic language.

PROCEDURES: combinatorics; a-priori analysis of the

experiment; Laplace’s rule.

CONCEPTS: favourable cases, expectation.

PROPOSITIONS: equiprobability; proportionality.

ARGUMENTS: systematic analysis of possibilities.

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Epistemic Meanings of Randomness

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3. PERCEPTIONS OF RANDOMNESS

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Research paradigms Some examples

Developmental stages Piaget, & Inhelder; Homeman, & Ross,;Green

Intuition and teaching Fischbein; Fischbein et al.; Konold et al.

Heuristics and biases Kahneman, Slovic, & Tversky; Bar- Hillel; Konold;

Martignon; Shaughnessy,;Serrano

Generating random sequences

Falk; Green; Wagenaar; Engel, & Sedlmeier;

Batanero, & Serrano

Comparative likelihood Falk; Green; Bar-Hillel, & Wagenaar; Toohey;

Chernoff

Modelling random experiments Eichler, & Vogel

Technological random experiments

Pratt; Pratt & Noss; Johnston-Wilder, & Pratt;

Paparistodemou; Noss, & Pratt; Cerulli,

Chioccariello, & Lemut

Analysis of own intuitions

Batanero, Arteaga & Ruiz

Developmental Stages and Intuitions

Piaget and Inhelder (1951) described developmental stages in probabilistic reasoning and predicted a mature comprehension of randomness at the formal operational stage.

Where will the next snowdrop fall?

However, probabilistic

reasoning does not always

develop spontaneously without

instruction (Fischbein, 1975) and

students’ recognition of

random distributions does not

improve with age (Green, 1989;

Engel, & Sedlmeier, 2005.

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People use specific heuristics to simplify uncertain

situations (Kahneman et al.,1982) . Other people do not

understand the purpose of probability (Konold, 1989).

Explanation: in probability counterintuitive results

abound even with basic concepts such as independence or

conditional probability (Borovcnik , & Peard, 1996); Borovcnik,

2014).

Later research identified the power of representation

formats (Gigerenzer, & Hoffrage 1995; Sedlmeier, 1999;

Martignon, & Wassner, 2002).

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Heuristics and Biases

Perception of Randomness

Generating tasks

Comparative

likelihood tasks (Chernoff)

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Which of the following sequences is the least likely to occur

from flipping a fair coin five times? Justify your response.

a. THTTT

b. THHTH

c. HHHTT

d. HTHTH

e. all four sequences are equally likely to occur

Working or Modelling with Technological Tools

Cerulli, Chiocchiariello, Lemut, 2005

Pratt, 2000; Johnston-Wilder, et al. 2008

Pratt& Noss, 2002

Paparistodemou, 2005; 2014

Lee, &

Lee,

2009

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Analysis of own Intuitions (Batanero et al.)

1. Subjects are given a generating task.

2. They are asked to analyse the data in the whole

classroom.

.

•Number of heads

•Number of runs

Obtaining

conclusions on their

own intuitions

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INVENTED SEQUENCE OF HEADS AND TAILS

H H T H T T T H H T H T H T T H H H T T

REAL COIN TOSSING SEQUENCE

T H T H T T H H T H H H T T T T T H T T

Analysis of own Intuitions

Subjects produce and interpret graphs of both

distributions.

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Perceptions of Randomness: a Summary

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Students’ Conceptions

Batanero, Gómez, Gea, & Contreras (2014)

- Randomness as opposed to cause.

- Randomness as lack of control.

- Randomness as equiprobability.

- Randomness as stable frequencies.

- Randomness as lack of model, etc.

- Any of these views is partly correct and can lead to the view of randomness as multiplicity of models.

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4. TEACHING AND LEARNING THE IDEA

OF RANDOMNESS

• Before 1970: Classical

view of probability

• “Modern mathematics”

era: axiomatic method

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The philosophical controversy about the meanings of

randomness and probability has also influenced teaching (Henry, 1997; Raoult, 2013).

Students are encouraged to perform random

experiments or simulations.

This approach connects probability and statistics.

Helps students face their probability

misconceptions (Biehler, 1997; 2003).

Tool in the teaching of modelling. (Engel & Vogel 2004; Batanero,

Biehler, Engel, Maxara, & Vogel, 2005)

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Frequentist approach

It is important to clarify the distinction between probability and frequency (Girard, 2001; Henry, 2001).

We never get the exact value of a probability but an estimate of the same.

We should also take into account one-off decisions, where a subjective approach to probability is preferable (Carranza, & Kuzniak, 2008).

The idea of updating previous information on the light of new data is closer to how people think (Devlin, 2014).

It may help to overcome many paradoxes especially those linked to conditional probabilities (Borovcnik, 2011).

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Some Precautions

The didactic situations may be difficult to be

reproduced (Brousseau, Brousseau, & Warfield, 2002).

Though simulation is vital to improve students’

probabilistic intuitions and in materialize probabilistic

problems, it does not provide the key about why the

problems are solved (Chaput, Girard, & Henry, 2011).

A genuine knowledge of probability can only be

achieved through the study of some probability theory.

However, the acquisition of such formal knowledge

should be gradual and supported by experience with

random experiments.

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Some Precautions

5. FINAL REFLECTIONS

Probability is the only reliable means we have to predict—and

plan for—the future, it plays a huge role in our lives, so we

cannot ignore it, and we must teach it to all future citizens

(Devlin (2014, p. ix).

Accepting this fact set some consequences for research in

probability education, which is quickly increasing today, but

still have open questions.

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A Didactic Approach to Randomness

Primary School

• Language of chance.

• Simple experiments, equiprobable outcomes, manipulative materials, representations.

Middle School

•Non-equiprobable outcomes, physical experiments.

•Observation of natural phenomena.

•Computers simulation, exploring microworlds.

High

School

• Properties of random sequences.

• Informal approaches to inference.

• Revising probabilities through new information.

• Mathematical models of probability. 33

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RESEARCH IN PROBABILITY EDUCATION:

SOME OPEN QUESTIONS

How can we use student's personal views of

randomness to develop adequate notions of

probability?

What fundamental stochastic ideas should be taught

today? Are Heitele’s (1975) and Burrill & Biehler’s

(2011) lists complete? Should we teach these ideas in

each curricular level?

How can probability and statistics be complemented

in the school curriculum?

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How can we take advantage of technological tools?

How can we increase the citizens’ probabilistic

competence to dealing with risk?

Do teachers have adequate knowledge to teach

probability in the different conceptions?

How should we complement the education of teachers?

How can we foster probability education research?

What theories and methods are useful for

understanding teaching and learning probability?

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RESEARCH IN PROBABILITY EDUCATION:

SOME OPEN QUESTIONS

CARMEN BATANERO

Univers idad de Granada , España

www.ugr.es /~ba tanero /

UNDERSTANDING

RANDOMNESS Challenges for Research and Teaching