Download probability literacy, statistical literacy, adult numeracy, quantitative

---- Final pre-publication draft ---To appear as: Gal, I. (2005). Towards 'probability literacy' for all citizens. In G. Jones
(ed.), Exploring probability in school: Challenges for teaching and learning
(pp. 43-71). Kluwer Academic Publishers
IDDO GAL
<[email protected]>
University of Haifa, Israel
CHAPTER 2:
TOWARDS "PROBABILITY LITERACY" FOR ALL
CITIZENS: BUILDING BLOCKS AND
INSTRUCTIONAL DILEMMAS
Probability is the very guide of life. (Cicero)
Chance favors the preared mind. (Louis Pasteur).
1. INTRODUCTION
What do we want students to learn about probability, and why do
we want them to learn that? Two reasons/answers are often provided
in the mathematics and statistics education literature. The first is that
probability is part of mathematics and statistics, fields of knowledge
that are important to learn in their own right, as part of modern
education. A variation on this answer is that learning of probability is
a foundation for learning more advanced subjects such as sampling
and statistical significance (Scheaffer, Watkins, & Landwehr, 1998) or
topics in other sciences. The second answer is that the learning of
probability is essential to help prepare students for life, since random
events and chance phenomena permeate our lives and environments
(Bennett, 1998; Beltrami, 1999; Everitt, 1999).
These two reasons for learning probability, which are driven by
internal and external considerations, respectively, are not mutually
exclusive; both have merit and should influence our thinking about the
content and process of education. This chapter is based, however, on
the belief that it is essential to place sufficient emphasis on issues that
are external to the structure of probability as a mathematical and
statistical topic. We have to reflect on the nature of the probabilityladen situations in the real world that adults may have to understand or
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Graham A. Jones (Ed.), Exploring probability in school: Challenges for teaching and
learning, 43—70. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.
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cope with, and on the implications for needed knowledge and
educational experiences. Attention to real-world demands should not
be the only factor influencing curricular planning or teachers'
practices, but it must be a part of the considerations that guide what
gets planned, taught, assessed, and valued in the classroom.
This chapter focuses on "probability literacy", the knowledge and
dispositions that students may need to develop to be considered
literate regarding real-world probabilistic matters. The chapter is
organized in three parts. Part one reviews models of adult literacy,
numeracy, and statistical literacy that define the terrain in which
knowledge of probability is situated. Part two discusses five basic
elements of probability-related knowledge and points to some
dispositions that are needed for adults to be able to effectively
interpret and engage real-world probabilistic situations. Part three
examines implications for instructional practice and research.
2. THE BIG PICTURE: ADULTS' LITERACY, NUMERACY AND
STATISTICAL LITERACY
Probability in Context
As argued above, one key justification for the teaching and learning
of probability in school is that chance phenomena permeate our lives
in multiple ways. Notions regarding probability, uncertainty, and risk
appear in various messages that adults encounter, such as when
receiving forecasts of medical, financial, or environmental risks from
the media, marketer, public officials, physicians, counsellors, or
research organizations. Both professionals and lay adults from all
walks of life have to interpret, react to, or cope with situations that
involve probabilistic elements or different levels of predicability or
unpredictability (Gigerenzer, Swijtink, Porter, Daston, Beatty, &
Kruger, 1989; Gal, 2000). Professionals and lay people at times also
have to generate estimates of the likelihood of certain events,
regardless of how much formal training in probability they have had.
That said, in the world, events either happen (sometimes in
different amounts or intensities), or do not happen. Examples are
getting a heart attack, rain tomorrow, winning a lottery, going
bankrupt, or a nuclear disaster. Probability is not a tangible
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characteristic of events, but rather a perception, whether expressed via
a formal mathematical notation or informal means, of the chance or
likelihood of occurrence of events. Such perceptions depend on the
interaction between factors operating in external situations and within
persons who face these situations. Consider the Insurance example:
"Insurance". You and your spouse just had your first baby. Your
mother reminds you that, being 30 years of age and a parent, it is
time to think about purchasing life insurance and upgrading
your medical insurance. Your financial resources are limited.
You are overweight and have had a slight heart murmur from
birth. Your doctor says that people in your weight category
"usually have at least a 50% higher risk for heart problems", and
that "this risk is even higher for people with a heart history".
Your mother recalls that you and your siblings had frequent ear
infections each winter, and says "I am quite sure your baby will
visit the pediatrician every other week". Your insurance agent
tells you that your monthly payment for a term life insurance
policy will be relatively low if you start now, but "surely double
if you join after age 40 or 50, given the statistics about health
problems in older ages". [Questions]: What will you do? What
are your chances of developing a real heart problem within the
next 10 years? 20 years? Will you seek more information about
future heart risks? Go on a diet? Will you increase your medical
insurance? Buy life insurance now, or wait?
Many considerations affect your thoughts about such a situation.
For example: your world knowledge (facts or assumptions, such as
about the causes of heart attacks), your personal dispositions (e.g.,
how averse you are to taking risks; how much you trust official
statistics), your interpretation of probability-related phrases (e.g., "at
least a 50% higher risk", "surely double"); or your ability to
understand, manipulate, or critically analyze the quantitative
information given or implied (e.g., what is the actual probability of
getting a heart attack for people with a "normal" weight? How much is
"50% higher" of that?)
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It follows that people's thinking and behavior in probabilistic
situations is affected by multiple knowledge bases and dispositions.
The remainder of this section discusses three related but separate
complex constructs: literacy, numeracy, and statistical literacy, that
provide an overarching framework within which elements of
probability literacy can later be placed.
Literacy
The term literacy has been traditionally associated with the level of
reading and writing skills people need for minimal functioning in
society. By association, the usage of “literacy” when paired with a
term denoting an area of human activity (e.g., "computer literacy")
may bring up an image of the minimal subset of basic skills expected
of all citizens in this area, as opposed to a more advanced set of skills
and knowledge that only some people may achieve. Yet, many authors
warn against a simplistic interpretation of what literacy means and
argue that under scrutiny this term becomes complex (Venezky,
1990). This situation also develops regarding probability literacy.
The term literacy, when used to describe people’s capacity for
goal-oriented behavior, suggests a broad cluster not only of factual
knowledge and certain formal and informal skills, but also of desired
beliefs and attitudes, habits of mind, and a critical perspective (Gal,
2002a). In the area of mathematics, conceptions of "mathematical
literacy" (Kilpatrick, 2001) or "quantitative literacy" (Steen, 2001)
extend the definitions of the mathematical knowledge desired of
school graduates (NCTM, 2000), in recognition of the complex nature
of the everyday situations adults have to understand and manage.
At the same time, recent years have seen the broadening of
conceptions about what it means to be "literate" in other functional
areas. Most relevant for the present chapter are the constructs "health
literacy" (Nutbeam, 2000) and "scientific literacy" (American
Association for the Advancement of Science, 1995; Shamos, 1995).
Authors writing on needed skills in these areas note that some
understanding of probability is a must, such as for making sense of
forecasts (e.g., of global warming), or understanding the idea of onetime and cumulative risks (e.g., due to engaging in unprotected sex).
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Adult Numeracy
The term numeracy is less commonly used than literacy, and so far
has gained only some consensus on meaning. One view equates
numeracy with basic computational skills, in the same way that
literacy is viewed by some as mastery of basic or minimal reading and
writing skills (Baker & Street, 1994). A much broader view of
numeracy, and the one advocated by this and other authors (e.g.,
Johnston, 1999), focuses on people’s capacity and propensity to
effectively and critically interact with the quantitative aspects of the
adult world. Gal (2000) defines numeracy as an aggregate of skills and
knowledge, dispositional factors (beliefs and attitudes, habits of
mind), and more general communication and problem-solving
capabilities, that individuals need in order to engage and effectively
manage numeracy situations.
Numeracy situations may involve numbers, quantitative or
quantifiable information, or visual or textual information that is based
on mathematical ideas or has embedded mathematical elements. Three
key types of numeracy situations are described below, involving
computations, interpretations, and decisions, and all of them are
relevant to a discussion of probability literacy.
Computational (or generative) situations require people to count,
quantify, compute, or otherwise manipulate numbers, quantities,
items, or visual elements, and eventually create (generate) new
numbers. An example is calculating the total price of products when
shopping, or estimating odds when playing a game of chance.
Interpretive situations demand that people make sense of messages
that may involve quantitative issues but do not require direct
manipulation of numbers or quantities. An example occurs when
reading, in a newspaper, a report of results from a recent poll or from
a medical experiment, possibly involving references to percentages,
random samples, or likelihood of certain events such as side effects.
The response expected in such a situation is usually the creation of an
opinion or judgment. However, opinions or judgments cannot
necessarily be classified as “right” or “wrong”, as with responses to
computational or generative tasks. Rather, these responses need to be
judged in terms of their reasonableness or the quality of the arguments
or evidence on which they are based.
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Decision-making situations demand that people determine a course
of action, typically in the presence of conflicting goals, constraints, or
uncertainty. Decisions have several subtypes, among them are
planning (the identification, design, and choice of an optimal way to
use resources; Secretary of Labor’s Commission on Achieving
Necessary Skills, 1991) and choice (the selection of one or more
options from given alternatives; Clemen & Gregory, 2000), but also
evaluation (Yates, 2001). Compared to responses to interpretive
situations, responses to decision making situations, such as in the
Insurance example above, have a larger subjective component since
they depend quite heavily on people's assumptions about future trends,
preferences, value systems, and judgments of probabilities.
Human judgment regarding the probability of occurrence of various
events has received considerable attention in the psychological
literature. It has been shown that judgments are affected by the context
in which they are made (Kahneman, Slovic, & Tversky, 1982;
Fischhoff, Bostrom, & Quadrel, 1993). Yates (2001) notes that a
judgment is a process supporting a decision but separate from it. For
example, a physician's belief that there is a 70% chance of patient X
having a disease Y is a judgment that supports the physician's decision
to treat this patient as if he suffers from disease Y. Yet, the actual
decision takes into account additional information, including
contextual information such as about the patient's overall condition,
the potential consequences of the decision, and so forth.
It follows that the distinction among generative, interpretive, and
decision-making situations becomes blurred when judgments of
probability are involved. One's opinion about the probability of an
event may result from an interpretive or subjective process but can be
based on computations or estimations (Cosmides & Tooby, 1996).
This blurriness should be kept in mind when considering the three
views of probability, classical, frequentist, and subjective, discussed
elsewhere in this book (see Batanero, Henry, & Parzysz).
In further reflecting on the need for probability literacy as part of
developing overall numeracy, it is important to note that occupational
demands seldom involve knowledge of probability computations.
Surveys of hundreds of employers in the USA (Packer, 1997) have
shown that the key statistical knowledge required, if any, in workplace
contexts includes familiarity with graphs and charts, understanding of
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variation, or familiarity with some descriptive statistics. Hence, it can
be speculated that for most adults, knowledge of probability is of
relevance primarily for functioning in personal, communal, and
societal realms, where situations require interpretation of probabilistic
statements, generation of probability judgments, or decision-making.
Statistical Literacy
Probability literacy is also closely linked to statistical literacy, a
notion that has emerged in the statistics and mathematics education
literature in light of the assumption that most adults will be
consumers, rather than producers, of statistical information. Wallman
(1993), in her presidential address to the American Statistical
Association, advocated the need to enhance the population's ability to
understand and critically evaluate statistical results that permeate daily
life, and to appreciate the contributions that statistical thinking can
make in public and private, professional and personal decisions.
According to Gal (2002a), statistical literacy refers to people's
ability to interpret, critically evaluate, and when relevant express their
opinions regarding statistical information, data-related arguments, or
stochastic phenomena. Gal further argues that statistically literate
behavior requires the joint activation of both cognitive and
dispositional components. The cognitive component involves five
knowledge bases: literacy skills, statistical knowledge (including also
some knowledge of probability, even if informal), mathematical
knowledge, contextual or world knowledge, and knowledge of critical
questions that have to be asked. The dispositional component involves
the presence of a critical stance, that is, willingness to adopt
questioning attitudes as well as certain beliefs, such as a belief in the
power of statistical processes, a belief in the self as capable of
statistical thinking, and a belief in the legitimacy of adopting a critical
perspective on information one receives from presumably "official"
sources or from experts.
Thinking in terms of a developmental sequence, Watson (1997)
described three levels that reflect increasing degrees of sophistication
in statistical literacy: basic understanding of probabilistic and
statistical terminology; understanding of statistical language and
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concepts when they are embedded in the context of wider social
discussion; and the ability to apply a questioning attitude to statistical
claims and arguments. These levels were supported by findings on the
performance of students at different ages on diverse tasks, including
probability-related tasks (Watson, & Callingham, 2003). The work of
Watson and Callingham also pointed to the importance of being able
to engage contexts and apply statistical knowledge in context as one
core aspect of students' emerging statistical literacy.
Overall, people need "probability literacy" to cope with a wide
range of real-world situations that involve interpretation or generation
of probabilistic messages as well as decision-making. However,
details of the probability-related knowledge and dispositions that may
comprise probability literacy have received relatively little explicit
attention in discussions of adults' literacy, numeracy, and statistical
literacy. The next section therefore takes a closer look at the elements
involved in probability literacy.
3. PROBABILITY LITERACY: KNOWLEDGE AND DISPOSITIONS
This section describes five key classes of knowledge and some
dispositions that are proposed as the building blocks of probability
literacy. These elements, listed in Table 1, follow the logic used by
Gal (2002a) in describing the construct of statistical literacy.
Several notes should be made about the proposed model. First, the
elements are listed in Table 1 separately for ease of presentation.
However, all elements are assumed to interact with each other in
complex ways during actual behavior or learning. This means that an
instructional focus only on one or two of the elements will not be
sufficient to develop "probability literate" behavior.
Second, dispositional elements, despite being of much importance, are
not discussed in detail in this chapter, given space limitations and
since some of the key ideas have been explicated in detail elsewhere
(Rutherford and Ahlgren, 1990; McLeod, 1992; Gal, 2000; 2002a;
2002b). Dispositions play a key role in how people think about
probabilistic information or act in situations that involve chance and
uncertainty, whether in the real world or in the classroom. A few
relevant examples are provided later on. Dispositions may also
influence students' willingness to learn more and further develop their
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probability literacy beyond what was possible during their initial
exposure to this topic in school (Gal, Ginsburg & Schau, 1997). The
fact that dispositional elements are not covered in detail in this chapter
should not be interpreted as if they are not worthy of attention from
curriculum designers, teachers, learners, and researchers.
Knowledge elements
1. Big ideas: Variation, Randomness, Independence, Predictability/Uncertainty.
2. Figuring probabilities: Ways to find or estimate the probability of events.
3. Language: The terms and methods used to communicate about chance.
4. Context: Understanding the role and implications of probabilistic issues and
messages in various contexts and in personal and public discourse.
5. Critical questions: Issues to reflect upon when dealing with probabilities.
Dispositional elements
1. Critical stance.
2. Beliefs and attitudes.
3. Personal sentiments regarding uncertainty and risk (e.g., risk aversion).
Table 1: Probability Literacy – building blocks
Finally, the five knowledge elements of the model of probability
literacy are described in broad strokes only because probability
literacy, just like numeracy or statistical literacy, is a dynamic and
relative construct. What constitutes a sufficient level of knowledge or
understanding in the area of probability cannot be defined in absolute
terms, and different cultures and life contexts pose diverse and
changing demands in this regard. Learners' age and background
impact on their world knowledge, ability to cope with abstract
concepts, or capacity and willingness to be critical of their own or of
others' thinking about probability, chance, and uncertainty.
Big Ideas
Familiarity with several foundational "big ideas", especially
randomness, independence, and variation, but also others, underlies
students' ability to understand the derivation, representation,
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interpretation, and implication of probabilistic statements (Moore,
1990; Snell, 1988; Peterson, 1998). Some aspects of these big ideas
can be represented by mathematical symbols or statistical terms, but
their essence cannot be fully captured by technical notations. Learners
must grasp the overall abstract nature of these ideas only intuitively.
Randomness is a slippery construct that has been debated by many
statisticians. According to Bennett (1998) one view is that randomness
is a property of an outcome, for example, whether the arrangement of
heads and tails after 10 coin tosses looks unordered or "random".
Another view is that randomness relates to the process by which an
arrangement came about (even if the arrangement itself looks orderly,
as when obtaining only heads in 10 tosses). A "random" process by
this view is one where events in the world occur without some
underlying deterministic cause or design that is fully predictable
(Beltrami, 1999). Given this elusiveness, it is not surprising that the
notion of randomness is sometimes left undefined or discussed only
informally by authors, teachers, and students alike (Dessart, 1989;
Green, 1989). Independence implies that events are unconnected and
one event cannot be predicted from another. Variation (and the need
for its quantification or reduction) is usually presented as the basic
motivation for any type of statistical investigation (Moore, 1990). In
the context of probability, variation underlies frequentist views of
probability, and can be extended to the idea that events and processes
vary in how certain we are that we can predict how they will unfold.
Randomness, independence, and variation are more complex ideas
than they may first seem, for several reasons. First, they have
complementary alter-egos: regularity, co-occurrence, and stability,
respectively; yet these are usually not discussed in teaching resources,
as if their presence is taken for granted. Second, each of the three pairs
of big ideas describes a separate continuum. Midlevels between the
end points can be envisioned, especially if the event being discussed is
not generated by devices such as dice or spinners. (e.g., the occurrence
of a thunderstorm is not a fully random event, nor is it a fully regular
event). Third, these three pairs, while separate, are also
interconnected, and their linkages have to be recognized by learners.
For example, the outcomes of a fully random process may vary more
than those of a process less affected by a random process.
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Notions of randomness, independence, and variation have to be
understood not only in their own right, but also as building blocks for
understanding a fourth pair of complementary big ideas, predictability
and uncertainty (and related notions of risk and confidence).
Predictability and uncertainty relate to the state of our overall
knowledge about the likelihood of a certain event (e.g., thunderstorm
tonight, winning a lottery). We may be able to describe the likelihood
of that event by a statement of probability (e.g., 10% chance, 1 in
1000). However, talking about an event's probability is not the same
as talking about its predictability or about our certainty regarding its
occurrence. An event's predictability depends on our assumptions
regarding the processes affecting the occurrence of that event and the
quality of the information we use to support estimates of probability.
When using random generating devices, a favorite of many
teachers, we may be able to state that there is a 50% chance of
obtaining heads in repeated tosses of a coin. This is a remarkable feat,
where we describe a long-range expected result despite our inability to
know for sure what will happen on each individual toss. We can
accomplish this because we can reflect on the underlying processes
and on the extent they involve randomness or independence, and we
can conduct experiments and examine stability in long-range patterns.
Events that are not created exclusively by random-generating
devices, however, are influenced by additional processes that we may
not fully know or understand. In such cases, even if we are willing to
make a statement of probability, we must also describe our level of
confidence in our predictions and make a statement of certainty.
Statements of certainty can manifest themselves informally in many
ways: A physician could say, in the Insurance example above, "I am
pretty sure that your heart murmur is not likely to develop into a
serious condition". Formally, notions of certainty are embodied in
statistical concepts such as "margin of error" or "level of
significance", whose understanding requires familiarity with random
sampling and sampling distributions, and hence with the underlying
big ideas of randomness, independence, variation, and certainty
(Beyth-Marom & Dekel, 1985, further discuss issues in defining
degrees of belief in formal and informal ways).
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Figuring Probabilities
Students have to be familiar with ways of finding the probability of
events, in order to understand probabilistic statements made by others,
or to generate estimates about the likelihood of events and
communicate with others about them. This is where the three views of
probability, classical, frequentist, and subjective, become useful (see
Batanero et al., this volume).
In school textbooks the classical view often takes precedence. It is
easy to use to establish familiarity with basic representations of
probability on the 0-1 scale, or with combinatorial computations
involving the probability of an intersection of events, such as
likelihood of getting "6" and "6" when two dice are rolled. Teachers
may further justify the emphasis on formal aspects of the classical or
frequentist approaches because they lay the foundation for learning
more advanced topics, such as sampling distributions, or behavior of
physical or chemical systems. Yet, outside the sciences, probabilities
are usually not computed in a simple and straightforward way, but
estimated or judged, and in ways that do not fit neatly only one of the
three views of probability. Usually, information from multiple sources
will be used, including nonprobabilistic information, and it will be
integrated through a rather complex judgment process.
As an example, imagine that you face the Insurance example given
earlier. To decide if to buy life and medical insurances, you have to
combine information from both statistical and nonstatistical sources
that are of varying quality: your physician's statements of probability
and certainty, statistical results that your physician mentions, your
mother's recollections, your own beliefs about your health, etc. Your
physician's statement of probability itself results from integration of
various pieces of information, such as long-range cumulative data
published in different medical journals, the physician's personal
assessment of your general nutrition and exercise habits, of your
family history, and so forth.
The above discussion suggests that as a minimum, people can be
expected to know that there are different ways to reach probabilistic
estimates, but also that estimates are often the result of the integration
of information from multiple sources. Also, they should be familiar
with the notion of evidence and understand that evidence (i.e.,
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quantitative and qualitative information) comes at different levels of
quality and that this quality can be evaluated or judged. Finally,
people should realize that statements of probability may have to be
accompanied by statements of certainty that describe one's level of
confidence in a probabilistic estimate. These expectations may appear
simple, even simplistic. Yet, their translation into classroom activities
requires creative planning as they suggest different instructional
trajectories than those used when basing instruction only on formal
notions of probability.
Some authors suggest that learners should also become familiar
with the more advanced idea of conditional probability, which
underlies Bayes' theorem. Utts (2003), for example, argues that all
learners should understand the difference between the two inverse
relationships, P(A|B) and P(B|A), that is, realizing that the probability
of being sick given that I have fever is different from the probability
that I have a fever given that I am sick. Familiarity with the essence of
Bayes' theorem is seen (e.g., Baron, 2000) as a gateway to avoiding
errors in one's own thinking, and enables one to detect erroneous
probabilistic judgments made by professionals and lay people alike,
who often confuse the two kinds of conditional probabilities.
Language
Numerous authors argue that students should understand the
"language of chance", that is, the diverse ways used to represent and
communicate about chance and probability (Rutherford, 1997;
Scheaffer et al, 1998; Steen, 2001). This section splits this general
expectation into two areas that have not been given much separate
attention in the educational literature: familiarity with terms and
phrases related to relevant abstract constructs, and with the various
ways to represent and talk about the likelihood of actual events.
Abstract constructs
As noted above, the domain of probability requires familiarity with
several complex concepts, especially variability, randomness,
independence, and (un)predictability and (un)certainty, but also
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chance, likelihood, or risk (Everitt, 1999). These abstract terms often
do not have crisp definitions that can be explained in simple language
or via references to tangible objects. Hence, coming to grips with their
meaning is not trivial and may be achieved only after a cumulative
process.
Words that describe these abstract concepts abound and are used in
a range of ways both inside and outside the classroom. For example,
learners may encounter the word “random” used with a specific
technical meaning in one setting ("We chose a random sample") but
also used informally in the media ("Random violence continued in the
streets…"). Not surprisingly, learners may attach to a word such as
"random" a diversity of meanings, including some not expected by
teachers (Gal, Mahoney & Moore, 1992). As with other areas in
mathematics education (Pimm, 1987; Gal, 1999), learners have to
become aware of the fact that the meanings of terms used in class are
often more constrained or precise than when they are used in everyday
speech. Terms used in a mathematics class may not necessarily carry
the semantic load implied in everyday discourse (Halliday, 1979).
This situation may affect learners' comprehension and increase the
chance for conflict or ambiguity in classroom talk, as potently
illustrated by Konold (1991). Thus, teachers should attend not only to
the extent to which they explain abstract concepts in clear language
and use them in a consistent way, but as well to students' ability to talk
(with understanding) about and with such terms.
Actual Probabilities
The likelihood of events can be represented quantitatively by
multiple systems, such as on a 0-1 scale, as fractions (e.g., 50/50),
percents, odds or ratios, and so forth, as well as graphically. Hence,
one basic expectation is that students understand the
interchangeability of different representations and feel comfortable
moving between them. (Of course, general mathematical knowledge is
needed here, such as for understanding the meaning of very small
numbers like "1 in 200,000" or "a chance of 0.0035", but also
regarding statements such as "50% higher risk" in the Insurance
example).
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However, attending only to quantitative representations is
insufficient, because probabilities may be communicated through
verbal phrases and because statements of certainty may be involved,
not just statements of probability. Consider the mother's words in the
Insurance example: The statement of probability is "…your baby will
visit the pediatrician every other week ". This simple description of
frequency is not on a 0-1 scale and is neither a fraction nor a percent
or ratio, and hence falls on the periphery of representations of
probability learned in school. Nonetheless, it does relate to likelihood.
(It could be translated into, say, "a 50% chance…to have to pay for a
medical visit once a week", but such a translation may not be sensible
in a real-world context). Further, the mother's statement of probability
is preceded by a statement of certainty, "I am quite sure...". This and
similar phrases ("I doubt that…") do not contain any numerical
elements, and hence cannot be translated to a numerical scale of any
kind based on formal probability theory. Yet, people are likely to use
such phrases, for example when they are unsure of their probability
estimates because their underlying information sources are shaky, or
because they had to integrate in a subjective way several pieces of
conflicting information.
The meanings that people attach to verbal phrases of probability
were studied in numerous contexts. Wallsten and colleagues (e.g.,
Wallsten, Budescu, Rapoport, Zwick, & Forsyth, 1986) have shown
that people vary in the way they interpret the likelihood conveyed by
words such as "likely", "probably", "surely", or by phrases that use
qualifiers (e.g., "very unlikely", "quite possibly"). Wallsten et al. have
shown that while some terms are interpreted quite narrowly (e.g.,
"almost certain" is equivalent to a probability of roughly 1.00-0.85),
people attach a wider range of probabilities to other, fuzzier terms
("good chance"). Further, people may interpret the probability
associated with the same phrase differently, depending on the base
rate of the phenomenon (Wallsten, Fillenbaum, & Cox, 1986).
It follows that students have to come to grips with the complexities
and vagueness inherent in using numerical and verbal means to
express both probabilities and certainties. Students should be given
opportunities to describe, orally and in writing, their thinking and
understanding about both likelihoods and certainties, and should see
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how others do that. This can help students to realize that people who
use the same "language of chance" may mean different things, and
such experiences can enhance students’ abilities to choose relevant
language (Beyth-Marom & Dekel, 1985; Konold, 1991).
Context
Being literate about probability-related matters requires that a
person develops some knowledge not only of relevant big ideas, ways
to figure probabilities, and the language of chance, but also of the role
of probabilistic processes and communications in the world.
Knowledge regarding context partially overlaps these former areas,
and is also related to the notion of "world knowledge" introduced by
Gal (2002a) as one of the five knowledge bases underlying statistical
literacy. Yet, the notion of "context knowledge" as used here
introduces specific expectations that people know both (a) what is the
role or impact of chance and randomness on different events and
processes, and (b) what are common areas or situations where notions
of chance and probability may come up in a person's life.
Knowledge pertaining to the context is necessary both from a
functional and an educational standpoint. Understanding that chance
and randomness do affect real-world events and processes in different
degrees enables people to anticipate that certain events will be more
predictable while others less so. Also, such knowledge underlies the
expectation introduced earlier that it is necessary for people and
organizations to have to make statements about the likelihood of
events, but also about the level of certainty behind such statements.
Understanding context is educationally important as it helps to explain
why there is a need to learn about probability or uncertainty in
different life circumstances. This is the basis for creating motivation
to study probability and for embedding the learning of it in socially
meaningful contexts.
Table 2 lists ten key areas from which useful examples can be drawn
to illustrate the occurrence and importance of randomness, variation,
probability, and risk. Other areas are of course possible, but the
guiding principle should be the same: to portray the omnipresence of
chance and randomness across the range of contexts which adults
encounter in different life roles, as workers, managers and planners,
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parents, consumers, patients, learners, citizens, environmentalists,
community activists, vacationers, sports enthusiasts, investors,
gamblers, and so forth. Many sources provide useful examples, such
as Gigerenzer et al. (1989), Paulos (1995), or the CHANCE project
archives (Snell, 2002).
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
the natural and physical world (e.g., weather, evolution)
technological processes (e.g., quality assurance, manufacturing)
human behavior (e.g., service encounters, sports, driving)
medicine, public health (e.g., genetic disorders, smoking-related risks)
justice and crime (e.g., matching of fingerprints or DNA)
finance and business (e.g., investment markets, insurance)
research and statistics (e.g., sampling, statistical inference)
public policy, forecasting (e.g., immunization)
games of chance, gambling and betting (e.g., dice, lotteries)
personal decisions (e.g., wearing seatbelts, college acceptance)
Table 2. Examples for contexts of probability literacy
Critical Questions
The last knowledge element in the probability literacy model
involves knowing what critical questions to ask when one encounters
a statement of probability or certainty, or when one has to generate a
probabilistic estimate. To illustrate the need for critical thinking about
probabilistic messages, consider the news item in Figure 1, which
appeared on the CNN website on September 23, 2003:
Studies: Lou Gehrig's rate higher in
Gulf War vets
WASHINGTON (AP) -- Veterans of the 1991 Persian
Gulf War were at least twice as likely to be diagnosed
with Lou Gehrig's disease as non-Gulf veterans or other
people younger than 45, according to two new studies.
Figure 1. Disease rates for war veterans
What does the news item in Figure 1 tell us about the probability of
getting the Lou Gehrig's disease? To make full sense of this seemingly
simple text, the reader has to do the following:
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1.
2.
3.
Notice that this article does make a claim about the
likelihood of a disease, yet in relative terms, that is,
"Twice as likely;" hence the actual probability is not
known (and for all we know could be negligible);
Notice that the statement of probability is made fuzzier
because of the modifier "at least" which can have two
different meanings: either that the figure provided
("twice") is rounded (i.e., the actual level is higher than
twice), or that the probability is an average of two
separate ratios, derived from comparisons to nongulf
veterans and to the general under-45 population in the two
studies mentioned; and
Be aware that contextual information not made explicit in
the text is needed to make sense of the information that is
given. The benefit of knowing that the disease is "twice as
likely" (without knowing the actual probability) does
become apparent, if the reader is aware of the debate as to
whether or not the rate of several illnesses among Gulf
war veterans was higher than usual, or caused by exposure
to toxic chemical agents.
The example above illustrates that readers and listeners cannot take
probabilistic statements for what they are, but rather have to be able to
ask a number of critical questions. The importance of being able to
ask critical questions, including about quantitative claims, has been
addressed by many authors. For example, Thistlewaite (1990) argues
that all learners should develop awareness of the need to question the
writer’s purpose, objectivity, or reasoning. Rutherford & Ahlgren
(1990) discuss the need for people to apply "critical-response” skills to
real-world statistical messages, and provide many useful examples.
Sources such as Utts (1996, 2003) and Gal (2002a) address the need
for learners to become familiar with elements of methodology that
affect the quality of a study's results (e.g., design and sampling,
measuring instruments), and with typical problems and biases that can
occur in reporting and interpreting statistical results (e.g., use of
misleading graphs, distinguishing correlation from causation, practical
meaning of "significant" differences, understanding what the existence
of a "margin of error" implies, etc).
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The range of issues that can be covered by critical questions in the
context of probability literacy is much wider than for questions
regarding statistical messages about results from surveys and
experimental research. Probabilistic claims can be based on results
from statistical studies, as illustrated in the Lou Gehrig example
above, hence all the issues listed in the last paragraph or in broader
discussions of criticalness become relevant. However, many other
issues are of concern, in light of the complexity of the big ideas
involved, the existence of multiple ways of deriving probabilities, the
role of subjective integration of information from multiple sources
(see the Insurance example), or the need to interpret both numerical
and diverse verbal descriptors of probability. In addition, probability
literacy is sometimes called for in generative situations or subsumed
as part of both personal and collective decision-making processes.
Given the above, a full discussion of questions that can help to
make sense and evaluate probabilistic claims is beyond the scope of
this chapter. Table 3 outlines key areas that such questions can address
when faced with probabilistic messages in interpretive contexts. These
questions relate to all four knowledge bases included in the model of
probability literacy, but additional critical questions could be called
upon, those usually associated with statistical literacy and with
numeracy (Gal, 2000; 2002a). It is not assumed that all the questions
listed in Table 3 will be relevant in all circumstances. Nonetheless,
trying to bring up all five areas to which questions relate as part of
instruction seems essential, even if it is done briefly or informally.
Finally, it is useful to note that in addition to the areas outlined in
Table 3, critical questions can pertain to "thinking errors" or judgment
biases related to estimations of probability. A large number of studies
have shown that people, including sometimes those with some training
in statistics, tend to estimate probabilities in inaccurate ways or think
about randomness, independence, variation, or risk in ways that
appear suboptimal or deviate from formal expectations (see Batanero
& Sanchez; Jones & Thornton; Watson; this volume)
While the reasons for such behaviors or phenomena are still being
studied or debated (Kahneman et al, 1982; Gigerenzer et al., 1999), it
would be useful for learners to at least be aware of such errors or
misconceptions, even if their exact mechanism is not always
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understood. Learners could be sensitized, for example, to variations in
how people act in or reason about some probabilistic situations, using
relevant phenomena such as the "gambler's fallacy" or "probability
matching" (Gal & Baron, 1996). Learners could also be made aware
of the negative influence of processes such as conservatism,
overconfidence, or availability, when judging probabilities of events.
Carefully selected examples in this regard can serve to highlight
formal principles as well as the presence of subjective processes, and
to show that people's beliefs and attitudes about random processes and
chance phenomena are complex and should not be seen as simple and
predictable.
1. Context. What is the nature of the domain about which a probabilistic
statement is being made? To what extent do the issues at hand involve
randomness, independence, variation, etc?
2. Source. Who is the source of a probabilistic claim (e.g., organization,
person), and what are his qualifications, expertise, characteristics, and motives?
3. Process. How did this source arrive at the claim being made? What types of
information sources were used (e.g., a "classical" analysis of equiprobable events;
frequentistic information or related data such as official statistics or results of
studies; subjective estimates)? What is the relevance of these data to the issue at
hand, and what is their quality? If multiple sources were used, how was the
information integrated or conflicts between data sources resolved?
4. Meaning of message. What is the meaning of the probabilistic statement
being made (numerical or verbal), and does it have to be translated or represented
in another way to be made clearer? To what exactly does the statement of
probability refer? (the issue of meaning may come up when a statement might
confuse P(A|B) and P(B|A), or when a source uses vague probability phrases)
5. Reflective interpretation. How should the message be interpreted? Should it
be questioned, given what is known about the context, the source, the derivation
process, and the clarity of the message's meaning? How reasonable are the
estimates made in light of one's world knowledge? Is it possible that one's own
assumptions and knowledge could be faulty? Or, is it possible that the probability
was over- or under-estimated by the source that generated it, due to self-serving
interests, hidden motives, need to err on the side of caution, risk aversion, etc.?
Table 3: Areas for critical questions-interpreting probability claims
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4. CONCLUSIONS AND IMPLICATIONS
Probability is intertwined into a wide range of real-world situations
and processes in both implicit and explicit ways. Adults need to be
able to effectively engage situations that require interpretation or
probabilistic messages, generation of probabilistic messages, or
decision making. The chapter outlined a model suggesting that
effective engagement with probability-laden situations requires the
presence of five knowledge-bases (regarding big ideas, figuring
probabilities, language, context, critical questions) as well as
supporting dispositions. These elements of probability literacy are
situated within a larger terrain of key competencies or essential skills,
already defined by broad and interconnected constructs such as
literacy, numeracy, and statistical literacy, as well as scientific literacy
and health literacy.
This proposed view of probability literacy can be seen as an
antithesis to the tendency in many school curricula to focus almost
exclusively on imparting formal knowledge pertaining to classical
and/or frequentist views of probability. For example, the Dictionary of
Cultural Literacy (Hirsch, Kett, & Trefil, 2002), which purports to
encompass all key concepts that a modern citizen should possess,
defines probability as:
A number between zero and one that shows how likely a certain event
is. Usually, probability is expressed as a ratio: the number of
experimental results that would produce the event divided by the
number of experimental results considered possible. Thus, the
probability of drawing the ten of clubs from an ordinary deck of cards
is one in fifty-two (1:52), or one fifty-second.
This conception is technically acceptable. Moore (1990) explains
that probability theory presents a body of mathematics that: "describes
chance in more detail than observation can hope to discover… an
impressive demonstration of the power of mathematics to deduce
extensive and unexpected results from simple assumptions" (p. 118).
Indeed, many authors derive very useful examples for instruction from
the seemingly simple situations involved in games of chance (e.g.,
Snell, 1988; Feldman & Morgan, 2003), and such games have
provided the impetus and context for the systematic study of
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probability by Pascal and Fermat in the 17th century (Batanero et al.,
this volume).
Nonetheless, definitions of probability such as that offered in the
Dictionary of Cultural Literacy can mislead educators, for they:
1. do not reflect the majority of actual uses of probability in
functional or cultural contexts,
2. do not embed a discussion of probability in the broader
context of adults' overall literacy, numeracy, and statistical
literacy,
3. can cause educators to assume that "teaching probability"
should involve mainly imparting technical knowledge of
probability-related computations.
Most everyday situations that require adults to activate their
probability literacy, that is, interpretive and decision making
situations, are not likely to require manipulation of probabilistic
information in pure numerical form in the range 0-1. The chance of
rain, for example, is described in weather forecasts using only verbal
or percent-based descriptors. Why, then, are computations with
probabilities in the range 0-1 emphasized in various textbooks and in
classroom tests? Historical reasons aside, they seem easy to explain
and enable teachers to perform demonstrations in class, using simple
random generating devices. Albert (2003) suggests that teachers and
test designers favor simple word problems such as "when rolling two
dice twice, what is the probability of obtaining on both rolls the result
'5'?” in part because their level of difficulty can be easily manipulated.
To counter the tendency to focus only on computational aspects of
probability, there is a need to attend to external demands in designing
curricula for developing probability literacy. The need to consider
functional demands in the real world when designing mathematics
instruction has been addressed by many authors and organizations,
and manifested in virtually all recent curricular frameworks (National
Council of Teachers of Mathematics, 2000; Packer, 1997; Rychen &
Salganic, 2003; Stein, 2000). James Rutherford, who has led Project
2061 of the American Association for the Advancement of Science,
stated (Rutherford, 1997) that since quantitative literacy is contextual,
the starting place for deciding what constitutes quantitative literacy is
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less in mathematics itself, than in the contexts where people are likely
to encounter the need for their mathematical skills and insights.
Balanced Instruction and Attention to Skill Transfer
At stake is whether we want to develop understanding of the
mathematics of probability per se, or contribute to students'
probability literacy (as envisioned in this chapter), and thereby to
students' broader numeracy or quantitative literacy (Steen, 2001). It
follows that the design of teaching for probability literacy, rather than
for technical or procedural knowledge, has to look at the big picture.
All five knowledge elements and the supporting dispositions outlined
here have to receive coordinated and balanced attention.
We have only partial understanding of teaching practices and
methods that can help students develop both the knowledge bases and
dispositions proposed in this chapter as building blocks of probability
literacy. Clearly, learning experiences and assessments have to be
couched in the context of meaningful situations, such as those related
to the areas listed in Table 2. An effort to develop any competency
that is contextual, however, including the development of probability
literacy, necessitates attention to the issue of skill transfer from inclass learning to situations outside the classroom. Extensive cognitive
research on learning (e.g., Lovett & Greenhouse, 2000) suggests that
students cannot cope well with new tasks (i.e., "transfer tasks") if they
do not get an opportunity to understand or practice key subskills.
Instructors cannot assume that students who have been exposed to
computational exercises and decontextualized tasks in class (e.g.,
results of using spinners or rolling dice) will be able to interpret,
reflect upon, and think critically about diverse probabilistic situations
and messages that they may encounter in real life.
As part of instruction, existing methods have to be extended and
new methods developed to go beyond computational procedures or
mathematical aspects of probability. Among other directions, the use
of computer technology has to be revisited so that computers are not
seen only as tools for running computations, conducting simulations,
or generating graphical displays of the probability of certain
outcomes. As much as such uses are important and interesting
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(Grinstead & Snell, 1997), there is a need to also touch upon big ideas,
the language of chance, the role of context in interpreting probability
messages or figures, and critical questions regarding probability
messages. This can only be achieved if students engage actual
probabilistic situations and messages that are messy and not as
structured as those teachers usually design for in-class exercises. For
example, students can be asked to search for explicit and implicit
statements of probability and statements of certainty in the Internet:
websites carrying media articles, bulletins of health organizations,
announcements of drug manufacturers, weather forecasts, press
releases of statistics agencies, or political statements. On the one hand,
students can be asked to use computer programs to generate numerical
or graphical representations. On the other hand, they can be asked to
write prose segments summarizing the results of experiments or
simulations, or to critique communicative messages that refer to
probability or certainty and have been found in various websites or
written by other students.
Once the focus of instruction goes beyond computational tasks and
learning experiences include some interpretive and judgment
situations, as well as communicative acts, assessments used by
teachers will have to be reconsidered (Jolliffe, this volume; Keeler &
Steinhorst, 2001). Assessments should examine not only
understanding of whatever principles were taught in class, but also
students' ability to apply their understanding to new tasks, as well as
whether students possess supporting dispositions. Finally, further
research efforts are warranted to increase the congruence between
visions presented in this chapter, teaching/learning processes, and the
extent to which learners can demonstrate probability literate behavior
outside the classroom.
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Iddo Gal, Ph.D., is a Senior Lecturer at the University of Haifa, Israel.