 Types of didactical
interventions and their characteristics
 It is
necessary before proceeding toward an optimal mathematical
didactical intervention in problem gambling to state its goals,
namely that of limiting the risk factors and result in a significant
desirable change in gamblers' behavior. These goals define the
optimality of such an intervention and distinguish it from a regular
mathematical intervention, where knowledge about and understanding
gambling mathematics are aimed. Regarding the setting in which such
an intervention could take place, there are three nonexclusive
options: in secondary to postsecondary public schools as optional
course or module attached to the probability/statistics courses,
within private companies or institutions dealing with gambling and
problem gambling, and as experimental interventions within the
problemgambling research.
 Although over the last two
decades, probability and statistics were present in the curricula of
most of the secondary schools as well as some 7^{th} and 8^{th}
grades worldwide, studies have indicated a decrease in the role of
probability and a greater focus on data processing at these
educational levels (Borovcnik, 2006). Among the reasons given for
this decrease there is the puritanistic view that probability is too
closely connected to games of chance, which are seen as plagues of
contemporaneous society even in the jurisdictions where they are
legal (Borovcnik, 2006). With this trend, probability theory came to
be taught in schools only because it is necessary to justify the
methods of inferential statistics. However, this principle is not
applied in all countries. The best example is Australia, where not
only do syllabi outline the role of probability in everyday life and
decision making, but teaching modules on the mathematics of gambling
have been implemented with success in the secondary schools (see
Peard, 2008). Some US states also implemented mathematics of
gambling within courses in public schools.
 Past studies on the impact of a
mathematical didactic intervention with gamblers, testing whether
learning about mathematics of gambling does change gambling
behavior, were mainly empirical (see Abbott & Volberg, 2000;
Gerstein et al., 1999; Hertwig et al., 2004;
Lambros & Delfabbro,
2007; Pelletier & Ladouceur, 2007;
Peard, 2008; Steenbergh et al.,
2004; Williams & Connolly, 2006). The experimental setup of those
studies was of two types with respect to the teaching module, which
was either assimilated with standard Probability Theory &
Mathematical Statistics courses taught in secondary and
postsecondary schools but including more applications from the
games of chance, or was designed and taught outside the curricula.
The content of most of the teaching modules fell within Introduction
to and Basics of Probability and Statistics, covering definition and
properties of probability, basics of descriptive and inferential
statistics, discrete random variables, expected value, classical
probability distributions, and central limit theorem. The modules
were packed with examples and applications from games of chance and
had lessons dedicated to demystifying mathematically the common
gambling fallacies.

 Results and
conclusions from the past and current interventions
 In the
literature on this matter, contradictory results have been published
and a clear conclusion has not yet been drawn. Some of these results
were declared by their authors as "paradoxical" or "unexpected," as
they did not confirm the expectation of a significant change in the
gambling behavior of the subjects. Thus, Hertwig et al. (2004) found
that students who received education on probability gambled on
lowodds events more than the students who did not know the actual
odds; Steenbergh et al. (2004) found that students who were taught
about and given warning about gambling fallacies and mathematical
expectation gained superior knowledge on these matters, but were
just as likely to play roulette as students who did not receive this
intervention; Williams & Connolly (2006) found that students who
received instruction on probability theory applied in gambling
demonstrated superior ability to calculate gambling odds, as well as
resistance to gambling fallacies, but this enhanced knowledge was
not associated with any decreases in actual gambling behavior. On
the other side, additional theoretical studies proved that
postsecondary statistics education developed critical thinking,
which also applied to gambling, and gamblers who get such education
tend to have significantly lower rates of problem gambling (Gray &
Mill, 1991; Gerstein et al., 1999; Abbot & Volberg, 2000).

 Conclusions of the conclusions
 These studies are problematic from
the standpoint of the experimental setup in three important areas:
sampling, evaluation, and testing of hypotheses. Regarding the
sampling, all studies were undertaken with groups of college
students, who are not representative of the gambling population with
respect to age as well as an assumed psychological profile whose
features could affect interpretation of the results. For example,
the fun/entertainment goal of gambling specific to this age group
might take precedence over other goals specific to different age
groups. With regard to evaluation, the studies were laboratorybased
and thus cannot reproduce realworld gambling activity. Filling out
a questionnaire on future intentions can neither substitute for nor
predict real actions. And finally, concerning the testing of the
hypotheses, two issues arise: first, the monitoring period should
not be the same for the entire sample group, but different for each
gambler according to his/her profile; second, there are constraints
induced by participation itself in the study, as gamblers are
apprised of the expectations of the study and thereby, their
gambling behavior might be influenced. Either of these issues may
provide an explanation for the contradictory results (Barboianu,
2013). Aside from these issues, two main questions arise:
 1. What mathematical knowledge
would an optimal teaching module contain, with respect to the
intended effect of limiting excessive gambling? In other words, what
is missing in the current didactic interventions?
 2. How important is the previous
mathematical background of the student for reaching the intended
effect? In other words, even if content and structure of the
teaching module are optimal, is it enough for the student to
understand and assimilate the mathematical facts presented, or is
mathematical thinking necessary—not only computational but also
conceptual inquiring, and highly critical, attainable only through
longterm previous mathematical education and experience?
 One thing is clear: If we teach
the mathematics of gambling with the goal of changing gambling
behavior, we must do it in a different manner, with respect to both
content and approach, from the customary methodology.

 Further
research
 Basing on a structural analysis of
the mathematical knowledge available for gamblers, as being attached
to the mathematical models of games of chance , we identified the epistemic knowledge related to
gambling mathematics and the act of modeling as one component missing in the regular
didactical interventions, and argued for its potential within an
optimized intervention (Barboianu, 2015). Considering the
functional models and not only the probabilistic & statistical
ones is another principle that we identified as having potential in
a didactical intervention, as creating for the student a
representation of the games as pure mathematical structures free
of risk factors (Barboianu, 2015, and forthcoming).
 Further research, both theoretical
and empirical, is necessary in various directions for establishing
the following:
  whether the learning principles
identified are practicably applicable to gamblers;
  what would be the optimal
content and structure of the teaching enhanced with such principles;
  whether the potential of this
nonstandard knowledge will actually manifest, given the various
levels of education of the gamblers.



 References
Abbott, M.W. & Volberg, R.A. (2000). Taking the Pulse on
Gambling and Problem Gambling in New Zealand: A Report on Phase One of
the 1999 National Prevalence Survey. Department of Internal Affairs,
Government of New Zealand.
Barboianu, C. (2013). The mathematical facts of games of
chance between exposure, teaching, and contribution to cognitive
therapies: Principles of an optimal mathematical intervention for
responsible gambling. Romanian
Journal of Experimental Applied Psychology, 4(3), pp. 2540.
Barboianu C. (2015).
Mathematical
models of games of chance: Epistemological taxonomy and potential in
problemgambling research.
UNLV Gaming Research & Review,
19(1).
Borovcnik, M. (2006). Probabilistic and
statistical thinking. In M. Perpinan, & M. A. Portabella (Eds.),
Proc. of the Fourth Congress of the European Society for Research in
Mathematics Education (pp. 484506). Sant Feliu de Guixols, Spain :
ERME
Gerstein, D., Volberg, R.A., Murphy, S.,
Toce, M., et al. (1999). Gambling impact and behavior study.
Report to the National Gambling Impact Study Commission. Chicago:
National Opinion Research Center at the University of Chicago.
Gray, T. & Mill, D.
(1991). Critical abilities, graduate education, and belief in
unsubstantiated phenomena. Canadian Journal of Behavioral Science,
22, pp. 162172.
Hertwig, R., Barron, G., Weber, E.U., Erev, I. (2004).
Decisions from experience and the effect of rare events in risky choice.
Psychological Science, 15 (8), pp. 534539.

Lambros, C. & Delfabbro, P. (2007). Numerical Reasoning Ability and
Irrational Beliefs in Problem Gambling. International Gambling
Studies, 7(2), pp. 157171.
Peard, R. (2008).
Teaching the Mathematics of Gambling to Reinforce Responsible Attitudes
towards Gambling. Retrieved from http://www.stat.auckland.ac.nz/~iase/publications/icme11/ICME11_TSG13_15P_peard.pdf
Pelletier, M., Ladouceur, R. (2007).
The effect of
knowledge of mathematics on gambling behaviours and erroneous
perceptions.
International Journal of Psychology,
42(2).
Steenbergh, T.A., Whelan, J.P, Meyers, A.W., May, R.K., & Floyd, K.
(2004). Impact of warning and brief intervention messages on knowledge
of gambling risk, irrational beliefs and behavior. International
Gambling Studies, 4 (1), pp. 316.
Williams, R.J., Connolly, D. (2006). Does learning about the mathematics
of gambling change gambling behavior? Psychology of Addictive
Behaviors, 20 (1), pp. 6268.

This entry should be cited as:
 Barboianu, C.
(2014).
DidacticalCognitive Contributions of
Mathematics to Problem Gambling.
Retrieved from
http://probability.infarom.ro/didactical.html.
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Gambling index
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