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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).
4
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).
5
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).
7
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).
8
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.
9
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).
10
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.
11
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.
15
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.
17
3. PERCEPTIONS OF RANDOMNESS
19
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.
20
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).
21
Heuristics and Biases
Perception of Randomness
Generating tasks
Comparative
likelihood tasks (Chernoff)
22
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
23
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
24
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
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.
27
4. TEACHING AND LEARNING THE IDEA
OF RANDOMNESS
• Before 1970: Classical
view of probability
• “Modern mathematics”
era: axiomatic method
28
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)
29
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).
30
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.
31
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.
32
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
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?
35
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?
36
RESEARCH IN PROBABILITY EDUCATION:
SOME OPEN QUESTIONS
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