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By Egan J Chernoff
Egan Chernoff is a former high school mathematics teacher with the Vancouver School Board.
Currently, he is an Assistant Professor at the University of Saskatchewan working with
prospective elementary, middle, and secondary mathematics teachers.
Given the theoretical, experimental, and
subjective interpretations of probability have
recently emerged as big ideas in worldwide
mathematics curricula (Jones, Langrall, &
Mooney,
2007),
understanding
that
probability and its associated nomenclature
is multivalent (i.e., has many interpretations)
is of timely importance. As such, the
purpose of this article is to demonstrate that
probability is multivalent, in general, and to
explore the multivalence of probability
terminology found in the teaching and
learning of probability, in specific.
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mathematical aspect and a foundational or
SKLORVRSKLFDO DVSHFW´ *LOOLHV S Alternatively stated, probability is open to
different meanings or interpretations;
probability is multivalent. +RZHYHU ³ZKile
an almost complete consensus and
agreement exists about the mathematics,
there is a wide divergence of opinions about
WKH SKLORVRSK\´ S The divergent
opinions are discussed in philosophy (e.g.,
Hacking, 1975, 2001), psychology (e.g.,
Coshmides & Tooby, 1996), mathematics
(e.g., Davis & Hersh, 1986), and
mathematics education (e.g., Shaughnessy,
1992). Further, literature found in different
fields is, unfortunately, wrought with
inconsistent terminology. For example,
alternative descriptors for the theoretical
interpretation of probability include a piori
and classical. The experimental and
subjective interpretations possess a wider
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variety of alternative descriptors, as seen
below in Table 1.
subjective
frequentist
Bayesian
a posteriori
intuitive
experimental
personal
empirical
individual
objective
epistemic
aleatory
belief-type
frequency-type
number 1
number 2
epistemologpropensity
ical
T able 1. Alternative terminology for
subjective and experimental probability
From the variety of names presented above,
classical, frequentist, and subjective are the
terms that have been adopted, by
acclamation, to represent the three different
philosophical interpretations probability
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mathematics classrooms. However, what is
actually meant by each of the three adopted
terms is open to debate. For example,
subjective probability for one individual
may mean something entirely different for
another individual. As such, it is imperative
that what is meant, or what can be meant, by
classical, frequentist, and subjective
probability is known by all individuals
involved with the teaching and learning of
probability, i.e., those who will use the
terms. In order to gain a deeper
understanding of what is (or can be) meant
by classical, frequentist, and subjective
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probability, the multivalence of probability
measurement
nomenclature
is
now
examined for concurrency.
The
interpretation
of
probability
predominantly known and used in teaching
and learning mathematics is the classical
interpretation. In fact, the use of the classical
interpretation is so prevalent, classical
probability and the probability taught in
schools, denoted here as school probability,
could be seen as one and the same: classical
probability seen as school probability and
school probability seen as classical
probability. Further, the utter dominance of
the classical interpretation being taught in
the mathematics classroom, especially senior
mathematics classrooms, means that
classical probability could also be seen as
formal probability, where ³SUREDELOLW\ LV
calculated precisely using the mathematical
ODZV RI SUREDELOLW\´ +DZNLQV .DSDGLD
1984, p. 349): classical probability seen as
formal probability and formal probability
seen as classical probability. Thus, and²I
think²syllogistically if arranged properly,
school probability can be seen as formal
probability and formal probability can be
seen as school probability. There is just one
problem: classical probability is different
from the formal aspect of probability and the
two should not be confused as one in the
same. Although the classical interpretation
of probability can be mistaken as formal
probability, it is important to remember that
classical probability represents one of the
(many) different philosophical aspects of
probability. As shown, classical probability
has the potential to be interpreted differently
by different individuals, but classical
probability is not the only multivalent term
used in the classroom.
While school probability may have once
been
dominated
by
the
classical
interpretation, there is increased use of the
subjective
and
frequentist
(i.e.,
experimental) interpretations of probability.
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As the use of the frequentist and subjective
interpretations in school probability
continues to grow, there are important
distinctions²between
and
within
interpretations²to note. The first of which
is to recognize that there is a fundamental
difference
between
frequentist
and
VXEMHFWLYH SUREDELOLW\ ³2Q WKH RQH VLGH LW
[i.e., probability] is statistical, concerning
itself with stochastic laws of chance
processes. On the other side it is
epistemological, dedicated to assessing
reasonable degrees of belief in propositions
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(Hacking, 1975, p. 12), respectively.
Hacking (2001) notes that until presented
with the distinction between frequentist and
subjective probabilities, the two are used
interchangeably. However, the distinction is
often difficult to make because the rules
used in the calculating of probabilities are
one and the same. Hacking (2001) further
notes that the dual classification, once
recognized, ³LV HVVHQWLDO«.for all clear
thinking about [and use of] SUREDELOLW\´ S
127).
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subjective probability because (1) the term
has finally made its way into the Western
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Common Curriculum F ramework (C C F) for
K-9 Mathematics (2006) and Common
Curriculum
F ramework
for
10-12
Mathematics (2008) and (2) the term is also
not immune from concurrent definitions. As
Gillies (2000) states:
The difficulty with this terminology is that
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probability include both the subjective
theory of probability, which identifies
probability with degree of belief, and the
logical theory, which identifies probability
with degree of rational belief. Thus,
subjective is used both as a general
classifier and for a specific theory. This is
surely unsatisfactory. (p. 19)
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Specific to the teaching and learning of
probability, terminology for the distinction
between the subjective theory (i.e.,
intrasubjective) and the logical theory (i.e.,
intersubjective) may not be a part of our
classrooms. As such, subjective probability
coexists as two specific theories of
probability and as a general classifier.
Consider, for example, the following
statement, ³:H WKLQN LW LV WLPHO\ IRU
researchers in mathematics education to
examine subjective probability and the way
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Langrall, & Mooney, 2007, p. 947).
Declaring, it is ti mely for researchers in
mathematics
education
to
examine
subjective probability uses subjective
probability as a general classifier, while
stating the way that students conceptualize it
uses subjective probability as a specific
theory. Chernoff (2008), in an analysis of
the state of probability measurement in
mathematics education, examined certain
subjective probability definitions showing
evidence for an implicit definition of the
term subjective. Subjective probability,
when used in the teaching and learning of
probability, was implicitly aligned with the
subjective theory of probability, and not as a
general classifier. Chernoff also concluded
that the implicit definition was closely
aligned with the personal theory of
subjective probability, and not the logical
theory.
Interestingly, the nomenclatural issues found
with subjective probability are also found
with frequentist probability. Similar to the
subjective interpretations,
µIUHTXHQWLVW¶
interpretations of probability include the
frequentist theory (i.e., relative frequency)
of probability and the propensity theory of
probability. As such, frequentist can also be
used as a general classifier, and for a
specific theory. However, specific to the
teaching and learning of probability,
terminology for the distinction between
frequentist and propensity theory may, also,
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not be a part of our classrooms. As a result,
the term frequentist coexists as two specific
theories of probability and as a general
classifier. That said, the burgeoning
connection between classical and frequentist
probability being utilized in the teaching and
learning of probability (Jones, Langrall, &
Mooney, 2007), suggests that frequentist
probability is implicitly recognized as the
frequentist theory, not the propensity theory,
nor as a general classifier.
While certain research (e.g., Chernoff, 2008)
attempts to refine probability measurement
nomenclature by proposing new terms for
adoption, there are a number of inherent
difficulties. For example, two options can be
taken into consideration. First, subjective
and frequentist probability may act as
general classifiers and the specific
theoretical notions of subjective and
frequentist probability can be further
refined. Second, subjective and frequentist
probability may describes specific theories
and the terminology of the general
classifiers can be changed. Unfortunately,
and despite adoption of either approach, the
change is not necessarily µbackwards
compatible¶.
Through an investigation of the concurrency
found in the multivalence of probability and
probability measurement nomenclature three
points have arisen. First, classical
probability does not represent the formal
aspect of probability. Second, the
distinctions of propensity and frequentist
theory within frequentist probability and
subjective and logical theory within
subjective probability do not yet have
terminological counterparts in the teaching
and learning of probability. Third, and as a
result of the second point, while frequentist
can mean both the propensity and frequency
theories, and a general classifier, frequentist
does mean the frequentist theory (i.e.,
experimental); and, similarly, while
subjective can mean both the personal and
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interpersonal theories, and a general
classifier, subjective probability does mean
personal subjective probability. Individuals
involved in using probability measurement
terminology must recognize that the terms
classical, frequentist, and subjective
probability are multivalent, and, further,
must be diligent in determining which
interpretation of the word is being employed
in communication when talking probability.
In short, to help meet the increased demands
of the probability classroom, one must know
the terms one uses.
4565457!58'
Alberta Education. (2006). Common curriculum framework for K-9 mathematics: Western and
northern Canadian protocol. Edmonton, AB: Alberta Education.
Alberta Education. (2008). Common curriculum framework for 10-12 mathematics: Western and
northern Canadian protocol. Edmonton, AB: Alberta Education.
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approximation. Philosophy of Mathematics Education Journal, 23.
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York: Harcourt Brace Jovanovich.
Gillies, D. (2000). Philosophical theories of probability. New York: Routledge.
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Hacking, I. (2001). An introduction to probability and inductive logic . Cambridge: Cambridge
University Press
+DZNLQV$6.DSDGLD5&KLOGUHQ¶VFRQFHSWLRQVRISUREDELOLW\²A psychological
and pedagogical review. Educational Studies in Mathematics, 15, 349-377.
Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to
classroom realties. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics
Teaching and Learning, (pp. 909-955). New York: Macmillan.
Shaughnessy, J. M. (1992). Research in probability and statistics. In D. A. Grouws (Ed.),
Handbook of research on mathematics teaching and learning, (pp. 465-494). New York:
Macmillan.
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