A journey of risk
Teaching concepts of risk within the
Mathematics classroom
Agenda
• How is risk defined in the “real world” and how it
can be explained in the classroom.
• The teaching cycle of risk and how it fits within
the New Zealand curriculum.
• Risk literacy and useful critical questions
• What concepts about risk do the students bring
into the classroom.
• What tasks can be used to uncover your students’
misconceptions and help teach risk.
• Really useful resources.
• Where to from here.
How risk is defined in the ‘real world’
• Risk is an awareness that we have developed to
help us understand and cope with the dangers
and perceived uncertainties of life where the
threat to an outcome is unknown.
• Everyday people make decisions that are based
on statistical information – investments,
insurance, product purchases, assessing tsunami
risk, measuring the effects of changes to
scheduling of transport services, political
decisions and medical procedures.
• The definition of risk often depends in the
industry or context in which it is being used.
How risk is defined in the ‘real world’
• One of the most common definitions of risk is as an
uncertainty based on historical information.
Risk can also be defined as –
• a hazard,
• a probability,
• a variability of a probability,
• the possibility of occurring a misfortune or loss,
• or as a threat that requires exposure and uncertainty.
• The fact that risk can have so many meanings often
causes problems in communication
How risk is defined in the ‘real world’
Risk analysis
• Expressed in mathematical terms a risky
event is one associated with a probability of
a loss of recourses like health, time, food or
money (Martigon, 2014).
• All risk decisions have the basic elements of
options, outcomes and uncertainties set in a
social context and framed in a language that
highlights some of the ways of looking at the
question (Fischoff and Kadvany, 2011).
How risk is defined in the ‘real world’
Risk analysis
• Risk analysis is the need to quantify risk
and is usually written in numerical terms.
This is what the Mathematics Curriculum
has introduced at Level 2.
• Risk analysis brings reason and scientific
deliberation to risk management.
How risk is defined in the ‘real world’
Risk as a feeling
• The intuitive reaction to danger, often automatic,
nonverbal and an experimental approach to
viewing risk.
• Even though decisions are based on statistical
information there is an emotional element that
means risk is subjective.
• People bring their world knowledge, personal
disposition and interpretation of probability
related statements as well as their ability to
understand, manipulate or critically analyse
information when it comes to risk.
How risk is defined in the ‘real world’
Risk as a feeling – the four biases
• Optimism bias – belief that one’s own risk is
lower than the risk faced by others sharing
the same behaviour.
• Risk denial – learnt cognitive skill,
strengthened by past and peer experiences
where new evidence only appears reliable if
it is consistent with a person’s belief.
How risk is defined in the ‘real world’
Risk as a feeling – the four biases
• Anchoring bias – the human tendency to
rely on one (often irrelevant) piece of
information when making decisions.
• Risk aversion – where losses loom larger
than gains. People will often take greater
gains to avoid little losses while neglecting
strategies to maximise long term gains.
The communication of risk
• The communication of risk is important in
society as a two way exchange of
information leads to the empowerment of
individuals through better understanding
and autonomy when making decisions.
• Poor communication could be lead to poor
decisions, denial of the right to know and to
participate in society and it can erode the
goodwill and trust society needs to operate.
The communication of risk
An example of poor communication and
understanding of risk
• In October 1995 there was a contraceptive pill scare
in Britain, when it was shown that there was a 100%
increase of potentially life-threatening blood clots
for women taking a new contraceptive, when in
reality the risk went from 1 in 7000 to 2 in 7000.
• The estimated fallout from the scare was an
estimated additional 13 000 abortions, an increase
of 800 pregnancies in girls under 16 and an
additional £46 million in health services as well as a
drop in confidence with oral contraception.
The communication of risk
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General misunderstandings
Can be due to several reasons – cognitive
limitations and biases or personal factors such as
culture or health.
The provider of information can choose the method
of communication that best suits its interests.
The time frame for the risk is not identified or not
the same period for the two groups being
compared.
When the reported risk is not the risk of the target
audience.
The communication of risk
General misunderstandings
Humans’ intuitive perception of risk can also cause
misunderstanding.
• People have an insensitivity to large numbers; the loss
of a few seems important. For example the loss of life
due to one off shootings seems more important than
the larger numbers lost to shootings in a war.
• Peoples’ insensitivity to probability, where the outcome
has a stronger effect on emotions than the numerical
value. For example the perception of risk is higher for
events that are uncontrollable, catastrophic or fatal.
The communication of risk
General misunderstandings
Humans’ intuitive perception of risk can also
cause misunderstanding.
• Lastly the numerator in a numerical
probability plays a part. 7 in 100 can be
seen as a better chance than 1 in 10.
The communication of risk
Verbal communication
The way in which language is used to convey risk
information.
• Has the potential to improve comprehension
of risk, capturing a person's emotions and
intuitions better than numbers.
• Potential weakness in the vagueness in the
variability of the individual interpretation of
probability statements.
• Usually framed in terms of a loss or a gain.
Verbal communication
Misunderstandings
• Positive framing has more of an effect in
persuading people to take risky treatment
options.
• Loss framing considers the potential losses from
not having a test and influences more people to
take the appropriate test than positive framing
would do.
• Loss framing also means consumers are more
likely to adopt high-risk products and influences
consumers positive mood about products.
Visual communication
A way to support risk communication
• Visual displays have the advantage to be able to
summarise data and reveal patterns that may go
undetected.
• Certain graphs lend them themselves to specific
tasks for example icons are used to show the
number of people affected in a population.
• Eppler and Aeschimann (2009) suggest
interactive visualisation may be better suited to
risk communication than print or text formats.
Visual communication
Misunderstandings
• Poor design and complexity or misinterpreted.
• They may discourage people from looking at
important details.
• Clear and comprehensible explanations with any
calculations need to accompany each graph.
Numerical communication
The use of numbers to quantify the magnitude of risk.
• Precise and verifiable with the ability to convert from
one format to another.
• The weakness is its lack of sensitivity to intuitions and
peoples’ level of numeracy.
• The way in which risks are expressed mathematically
can make a difference in the way risk is interpreted.
Numerical communication definitions
Numerical communication definitions
• Baseline risk – numerical information that
does not include the treatment or behaviour
in question. In practice it is hard to find and
depends on the population that is being
studied.
• Absolute risk – Looks at one category and
uses the same formula as probability, where
the number with the trait is divided by the
total in the category.
Numerical communication definitions
• Relative risk – the ratio of the absolute risks for
two groups, one being the group with the
characteristic of interest and the other being the
baseline or comparison group. Shows how much
bigger or small one risk is relative to another and
often expressed as a multiple.
• Natural frequencies – presented as large counted
events classified into groups. Easier to interpret
as this is the traditional way people process
numerical information.
Numerical communication definitions
• Odds ratio – When two odds are compared relative
to each other. Analytically it is easier to work with
than relative risk.
• Numbers needed to treat – The number of people
necessary to treat in order to prevent one more
unwanted outcome.
• Risk reduction – how many fewer people with the
desired trait there are with the treatment than
those without the treatment.
Numerical communication
Misunderstandings
• Numeracy levels of the target audience, for
example interchanging between fractions and
percentage formats.
• Excluding information about the baseline
(reference class). Humans often make decisions
without the baseline information and can be
avoided when natural frequencies are used.
• Humans have difficulty understanding small
probabilities due to the rarity in which we
experience them.
Numerical communication
Misunderstandings
• Relative risk is used when persuasion is the
goal.
• When relative risk is used the reference class
may be different to the intended target.
Confusion could be avoided with a mixture
of absolute risk and natural frequencies
along side relative risk.
• Risk from dramatic events tend to be
overestimated while risk from undramatic
events tend to be underestimated.
Risk in the classroom
• Risk is crucial in order for us to participate in the
modern world, therefore training young students
in the perception of risk has become fundamental
in modern society.
• Traditional probability instruction in school has a
tendency to be based on the probability axioms
and mathematical calculations.
• For the majority of us, the situations that
necessitate us to draw on probability knowledge
will be those requiring judgements or
interpretation, not calculation .
The teaching cycle of risk
The teaching cycle of risk
• Risk intuitions – preconceived ideas and bias that
people bring with them about risk and probability.
• Presentation of risk communication – all the
different formats of risk communication that a person
must be familiar with.
• Interpreting risk communication – understanding
where the numbers come from. Included is the
ability to change the format of how risk is presented.
The teaching cycle of risk
• Conversing with risk communication – being able
to explain risk in different formats so that the
receiver of information with very little
understanding of risk communication
understands the harms and benefits without
being mislead. This is the hard part of the cycle to
accomplish.
• Reasoning with risk – asking relevant risk literacy
questions (to come) about risk. This is a higher
level thinking and leads to new intuitions about
risk.
How risk fits into the New Zealand Curriculum
Achievement standard S7-3
Calculates and interprets risk, selects baseline
group, and calculates and interprets relative risk
and writes a news clip reporting on findings.
• In a media article with text and/or table,
identifies absolute risk, baseline group and
relative risk, and for relative risk, identifies the
two groups being compared;
• Identifies missing information and justifies why it
is important to include this information; identifies
whether risk applies personally and why.
How risk fits into the New Zealand Curriculum
• Currently the NZC is assessing at the
interpretation of risk communication stage of the
cycle on a small area of risk communication.
• While we have made great strides in teaching
students about the numbers behind risk
communication, the assessment of interpretation
of risk communication has not been linked to
student’s intuitions or other formats of risk
communication.
How risk fits into the New Zealand Curriculum
There are two aspects of the curriculum that
help build students’ understanding of risk
concepts.
• Probability – with the importance on
interpretation and evaluation of the language
including independence and conditional
probabilities when stated in a social context.
• Numeracy skills – a key feature in risk
quantification and communication.
What should be taught to help the
understanding of risk?
It is important to teach students to translate
probabilities into natural frequencies; especially
conditional probabilities.
• This is because the size of harms and benefits
become more transparent when expressed as
natural frequencies.
Students need to be made aware there are two
types of errors.
Risk literacy
A branch of Statistical literacy and in particular
probability literacy.
• The minimum components of being risk
literate are the main concepts like absolute
risk and learning to live with uncertainty
rather than more advanced topics such as
variability.
• People need to recognise that there is no
guarantee of zero risk, only risk that is more
or less acceptable.
Risk literacy
Probability literacy
Dispositional elements
Knowledge elements
• Critical stance
• Beliefs and attitudes
• Personal attitudes regarding
uncertainty and risk
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Variation
Randomness
Independence
Uncertainty
Figuring probabilities
Language
Context
Critical questions
Gal (2002)
Critical questions
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Risk of what?
The outcome to which the risk refers.
Size matters and it should be expressed in absolute
terms or the baseline information should be given.
There are two types of errors – false positives and
false negatives.
What is the frame?
Time frames such as the next ten years are easier
to understand than lifetime risk.
Wording.
Does it apply to me?
Risk of What?
• What is the baseline?
• Has information about the baseline been
included?
• Are any harms and benefits being expressed in
the same format?
• Is the risk in relative risk or absolute risk form?
• Can the be shown in different formats - visual
and numerical?
What is the frame?
• Is the time frame stated?
• Are the time frames for the two risks I am
comparing the same?
• What is the framing (wording) of the
question?
• Is it a positive or negative frame? Is it a loss or
gain frame?
Does it apply to me?
• To whom does the risk apply?
• Do I share the same characteristics as
these people?
Student reasoning and conceptions
Availability Heuristic
Judgement is created on the availability or
recall of information in the memory.
• Leads to sensible use of own contextual
knowledge to judge reasons for likelihoods
when the likelihoods of risks are given.
• On the other hand it could lead to errors in
judgement when the likelihood of risks are
estimated.
Student reasoning and conceptions
Critical questions to ask when judging risk
information
Risk of what?
• The size of risk depends on the baseline was
not taken into account when judging risk but
mentioned when asked how to determine
risk.
• Appropriate reasoning about risk
communication depends on the format
presented.
Student reasoning and conceptions
Critical questions to ask when judging risk
information
What is the frame?
• Time frame is not considered when it was
not given.
• Larger numbers influence a preference for a
positive frame.
• A preference for risk expressed as a harm or
benefit is not always evident in students.
Student reasoning and conceptions
Critical questions to ask when judging risk
information
Does it apply to me?
• The relevant sample space is not taken
into account unless it is given to the
students when it comes to judgements
about risk.
Student reasoning and conceptions
Reasoning numerical with risk
Estimating risk - Sense of the size of risk.
• Considered the content and
consequence.
• Developed a sense of numbers associated
with high and low risk.
• Personal control ideas tended to lead to
disregard of universal sample space
Student reasoning and conceptions
Reasoning numerical with risk
Calculating risk
• Natural frequencies led to appropriate
calculations.
• Proportions led to inappropriate calculations.
Language of risk - Everyday language vs. probability
language.
• Some perceived risk as having a negative
connotation.
Student reasoning and conceptions
Sample space
Universal sample space – All population units
• Tended to disregard when assessing risk.
Conditional sample space – Subset of population
• Used when thinking about risk applying to me.
• Conditioning on the event – where the students
conditioned the event on the population or it was
assumed that the event was already underway.
Student reasoning and conceptions
Sample space - Conditioning on the event
Teaching activities
Define risk - what it might uncover
• The negative, everyday language associated
with risk and how students define risk.
• Personal control over risk. Those that
perceive that they have control see the risk as
small.
• Critical questions. The risk of what? (The size
of the risk). Does it apply to me? The time
frame of the risk.
Teaching activities
Rank these risks
• On the right is a list
of events that
people could die
from.
• Put the events in
order from one to
ten, where one
carries the most risk
and 10 the least
risk.
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Falling
Firearm assault
Motor cycle accident
Motor vehicle accident
Pedestrian accident
Stroke
Smoking
Suicide
Natural forces (quakes etc.)
Drowning
Teaching activities
My study
National Safety
council (USA)
Falling
9
5
Firearm assault
7
6
Motor cycle accident
4
8
Motor vehicle accident
5
3
Pedestrian accident
3
9
Stroke
6
2
Smoking
8
1
Suicide
1
4
Natural forces (quakes, etc.)
10
10
Drowning
2
7
Teaching activities
Rank these risks - What it might uncover
• Availability heuristic – What has just
happened recently in the news and to
friends and family of your students may
influence how risk is interpreted.
• Critical questions – Does it apply to me?
Did you base the ranking of risk on your own
risk, the risk of those in your town/city, the
risk of New Zealanders or the risk of
everyone in the world?
Teaching activities
Visual representations of risk - Graphical
Teaching activities
What it might uncover
• Unfamiliar format for students where they are
asked to interpret what the different lines of R
might mean.
• Students may use the availability heuristic for
topics that are unfamiliar to them.
• For interactive graphs go to
understandinguncertainty.org/view/animations
Teaching activities
Visual representations of risk - Pictorial
Eating bacon is bad for you!
Unit of work - uses 100 picture frame to
evaluate risk
• Only a snapshot is provided for you, the rest
is at motivate.maths.org or visit
OCarrollMaths on the risk page for the link.
• Alternative is understandinguncertainity.org
for interactive applications.
• PowerPoint
Teaching activities
Numerical risk representations
Version One - Proportions
• The probability that a woman of age 40 has
breast cancer is about 1 percent. If she has
breast cancer, the probability she tests positive
on a screening mammogram is 90 percent. If she
does not have breast cancer, the probability that
she nevertheless tests positive is 9 percent. What
are the chances that a woman who tests positive
actually has breast cancer?
Teaching activities
Numerical risk representations
Version Two – Natural frequencies
• Eight out of every 1000 women have breast
cancer. Of these eight woman with breast cancer,
7 will have a positive mammogram. Of the
remaining 992 woman who don’t have breast
cancer, some 70 will have a positive
mammogram. Imagine a sample of 1000 women
have a positive mammogram. How many of
these women actually have breast cancer?
Teaching activities
What it might uncover
• Students provided with numerical information
give the impression that they were able to retain
information about the relevant sample space
whilst discussing the problem, thereby arriving at
the correct answer.
• Try Positive test activity at motivate maths(same
site as the bacon activity. Only a snapshot has
been given to you. Or visit O’Carroll Maths for
the link.
• Helpful with converting between natural
frequencies and probability.
Useful Resources
• Anything by Gerd Gigerenzer or David
Spiegelhalter. They are two of the leading
experts on risk. They have written several books
about risk.
• The Risk activities booklet has useful websites
with class activities, YouTube videos and
background information on risk.
• I am slowly adding more Risk activities to
O’Carroll Maths. Check back every now and then.
Where to from here?
Implications for teaching
• Teachers need to aware that students come into
the classroom with their own intuitions –
sometimes beneficial but can be detrimental.
• Teachers need to be more familiar with risk
communication and critical questions.
• It is desirable for learners to experience risk
information in formation in several formats,
therefore teachers need to familiarise themselves
with different formats of risk.
Where to from here?
Developing the probability curriculum
• Probability is often the topic that is left to
last, squashed into a couple of lessons,
tacked onto the end of statistics or
completely left out as it might appear that
there is little purpose for the study of it.
• Reasoning with risk is the reason we teach
probability and is a main driving force
behind how our society works.
Where to from here?
Developing the probability curriculum
• Thinking about risk needs to start earlier for
students than in the curriculum at Level 2.
• Risk need to be the driving force behind
teaching probability. It is the real life
application for the students (insurance,
investments, medicine and product
purchases) – unlike coloured balls in a bag.