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EDMA310 – Assessment Task 1
Literature Synthesis – Probability: Experimental and Theoretical
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According to Truran (1994) there are three ways in which people think about probability,
those being subjective, experimental (empirical), and symmetric probability. In other cases
experimental is referred to as empirical, and in most other cases symmetric is named
theoretical (Barnes, 1998; Chick, 1998; & Meagher, 2012). The theoretical approach to
probability is centred upon a “logical analysis of the experiment” (Van de Walle, Karp, &
Bay-Williams, 2010, p. 462), that is, assessing the likelihood of outcomes without requiring
experimental data. Subjective probability identifies and relies on the beliefs in which
students have developed from past experiences regarding chance events, many of which are
not supported by theoretical analysis. Finally, experimental probability refers to probabilistic
outcomes determined through the process of trialling in chance events (Chick, 1998). It is
through the method of experimental probability that students are most able to mature from
their preconceived beliefs and develop towards seeing the link to theoretical probability
(Barnes, 1998). More so, experimental probability has been defined as estimated probability
based on data (Konold, et al., 2011).
This is not the only link to be found between experimental and theoretical probability. The
Law of Large Numbers is also vital in showing how each approach reflects the other. In
undertaking probabilistic problem solving through the experimental method, as the number
of trials increases the overall data becomes a closer representative of the theoretical
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analysis (Konold, et al., 2011). To assess experimental and theoretical probability in the case
of one specific chance event they will often show varying results, however if the importance
of the law of large numbers is understood and adhered to, the results of the theoretical
analysis and empirical data will usually reflect each other. Meagher (2012) states the
relationship between these two methods of probability simply but well, illuminating that “in
some cases an experiment can be used to confirm a theory and in other instances it can be
used to develop a theory” (p.14).
Although the concept of probability has been part of the curriculum for some time, it still
does not have the emphasis placed upon its importance that it should. Additionally it is only
in recent years that research has been made into awareness and development of the
connection between experimental and theoretical probability (Ireland, & Watson, 2009).
This presents a problem in that current teachers may not be ensuring that students are
aware of the importance of probability as an overall subject, or even more so experimental
and theoretical approaches. It is important that students see the link to real life situations,
as it is often a misconception that chance is not important which leaves the student without
purpose for exploring meaningfully or being able to apply their knowledge acquired to reallife situations (Barnes, 1998; Chick, 2006; Smith & Taylor, 1998).
As briefly mentioned previously it is important that students are exposed to experimental
and theoretical probability problems in order for students to discover the truthful outcomes
and hence lessen their reliance on their subjective beliefs. Barnes (1998) and Shaugnessy
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(1993) both make mention that it is a long and hard task to change students’ subjective
beliefs, yet endeavouring this through experimental activities is the favourable strategy.
An interesting finding between statements by Shaugnessy (1993) and Barnes (1998) was
discovered in regards to the awareness fallacy and the probability of outcomes. Often due
to subjective beliefs/misconceptions such as the awareness fallacy, children will state that
one outcome is more likely than another in an equal outcomes situation, whereas
Shaugnessy’s (1993) study shows a change in this as students get older (Barnes, 1998).
Shaugnessy (1993) found a misconception in his study on older students to be that many
believed “that all outcomes in any probability experiment are equally likely” (p.245), which
is inaccurate, therefore the view upon likeliness of outcomes has changed to each extreme
between the younger and older children.
The awareness fallacy mentioned concerns the misconception that students will think a
certain outcome is more probable due to it having been drawn to their attention in
experiences, or that it is less likely as they have paid particular attention on how often the
outcome does not occur (Barnes, 1998). Chick (1998) addresses this as the recency effect or
availability heuristic, although she more specifically links it with experiences having occurred
recently.
The Law of Large numbers introduced earlier presents a misconception in itself. A number
of experts state the common problem that people believe only a small amount of trials will
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provide a realistic representation or insight into the ‘big picture’, or an accurate comparison
to the theoretical analysis (Konold et al., 2011; Barnes, 1998; & Shaugnessy, 1993). Tversky
and Kahneman (1971) refer to this as the “belief in the law of small numbers” (as cited by
Konold, et al., 2011). Ireland and Watson (2009) addressed this to be considerably due to
students’ misconception of ‘fairness’ entailing the expected results should be achieved in
the short-term.
Another misconception is that chance has memory, as opposed to Van de Walle, Karp, and
Bay-Williams’ “chance has no memory” (2010, p.456). Many people believe that
independent outcomes are often dependent on outcomes prior to it, yet this is not an
accurate view. Named the ‘gambler’s fallacy’, a simpler explanation is that people often
have a false belief that if an outcome has occurred frequently, then the other is more likely
to occur soon despite its theoretical probability (Barnes, 1998). Relative to this
misconception people may fail to acknowledge the possibility that the results will vary
(irrespective of prior outcomes) and may not reflect the theoretical probability in a small
number of trials. Barnes (1998) refers to this as “misconceptions about randomness” (p.18).
In Green’s (1983a, 1983b) study of adolescents it was found that students probabilistic,
mathematical language was not adequate for what was expected, demonstrating that the
language of probability should be implemented and encouraged from the beginning and
continually throughout the students education (as cited in Shaugnessy, 1993). In the
Australian Curriculum of Mathematics the first aspect of probability is to develop the
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students’ language, similarly Konold, et al. (2011) highlight that it may be important to
define and encourage the students use of the language ‘experimental’ and ‘theoretical’
probability in the initial stages of teaching the approaches.
Subsequently, to teach theoretical probability (and somewhat experimental probability)
the specified misconceptions will then need to be addressed. The very first step would be to
provide the students with an activity or prompt discussion about a probabilistic statement in
order to gauge what knowledge and beliefs students hold in regards to probability (Barnes,
1998). It would then be appropriate for the students to learn about the approaches through
inferring and engaging in experimental problems. Shaugnessy (1993) suggests that in
teaching students the concepts of experimental and theoretical probability we should
provide them with “probability tasks that have been used in research” (p.247).
In Meagher’s (2012) research he proposes a series of experimental and theoretical
approaches. Firstly he advises activities based on theories that will be supported by the
experimental evidence, following he encourages to challenge the students with an activity
where the evidence may not so obviously support the theory in order to promote deeper
thinking and inferring strategies. Lastly an activity where a theory has to be developed from
the evidence gathered via completing the experimental task. This succession will scaffold
and assist students’ exploration and development of understanding of the relationship
between experimental and theoretical probability (Meagher, 2012).
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Many others sources provide examples of games in order to introduce and educate
students on the concepts of experimental and theoretical probability, such as Barnes (1998,
p.19) recommends “Dice differences” (Lovitt, & Lowe, 1993) as a beginning game for
students to begin exploring with (Chick, 1998; Smith, & Taylor, 1998).
In addition to these points, Shaugnessy writes of the value of teaching probability “in an
open-ended problem-solving environment in small groups, with heavy use of simulations”
(1993, p.247).
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References
Australian Curriculum, Assessment, and Reporting Authority (ACARA). (n.d). Mathematics:
Foundation to Year 10 Curriculum.
Barnes, M. (1998). Dealing with Misconceptions About Probability. Reflections, 20(1), 60 –
63.
Chick, H. L. (2006). Big Ideas in Chance. Paper presented at the Mathematics – the way
forward: 43rd conference of the Mathematical Association of Victoria, Brunswick,
Victoria.
Ireland, S., & Watson, J. (2009). Building a connection between experimental and theoretical
aspects of probability. International Electronic Journal of Mathematics Education,
4(3):339 – 370.
Konold, C., Madden, S., Pollatsek, A., Pfannkuch, M., Wild, C., Ziedens, I., Finzer, W., Horton,
N. J., & Kazak, S. (2011). Conceptual Challenges in Coordinating Theoretical and Datacentred Estimates of Probability. Mathematical Thinking and Learning, 13(1-2), 68 –
86.
Meagher, M. (2012). The Interplay Between Theoretical and Experimental Probability:
Beyond “Sample Size Matters”. Ohio Journal of School Mathematics, 65 (), 14 – 20.
Shaugnessy, M. J. (1993). Probability and Statistics. The Mathematics Teacher, 86(3), 244 –
248.
Smith, R., & Taylor, S. (1998). Dealing with Chance: Prep to Year 4. In J. Gough, & J. Mousely.
(Eds), Mathematics: exploring all angles (Proceedings of the 35th Annual Conference
of the Mathematical Association of Victoria, pp. 401 – 408). Brunswick, Victoria:
Mathematical Association of Victoria.
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Truran, J. (1994). What is the Probability of…?. The Australian Mathematics Teacher, 50(3),
28 – 29.
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Information in Learning Trajectory sourced from:
Australian Curriculum, Assessment, and Reporting Authority (ACARA). (n.d). Mathematics:
Foundation to Year 10 Curriculum.
National Council of Teachers of Mathematics (NCTM). (2012). Principles and Standards for
School Mathematics: Data Analysis and Probability. VA: NCTM
Victorian Curriculum and Assessment Authority, State Government of Victoria. (2007).
Mathematics Domain. Victoria: Author.
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