a journey of risk teaching concepts of risk within the mathematics classroom

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.


• 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


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).


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.


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.


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.


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.


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.


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.


General misunderstandingsHumans’ 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.


General misunderstandingsHumans’ 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.

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


• 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.


• 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.


• 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.


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.


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 S7-3Calculates 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.


• 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.


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

• Variation• Randomness• Independence• Uncertainty• Figuring probabilities• Language• Context• Critical questions

Gal (2002)

Critical questionsRisk 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 HeuristicJudgement 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.


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.



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.



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.


Reasoning numerical with riskEstimating 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


Reasoning numerical with riskCalculating 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.


Sample spaceUniversal 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.


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.

• Falling• Firearm assault• Motor cycle accident• Motor vehicle accident• Pedestrian accident• Stroke• Smoking• Suicide• Natural forces (quakes etc.)• Drowning

Teaching activitiesMy 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

http://understandinguncertainty.org/view/animations

Teaching activities

Visual representations of risk - PictorialEating 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

https://motivate.maths.org/content/MathsHealth/Risk

Teaching activities

Numerical risk representationsVersion 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 representationsVersion 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 activitiesWhat 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.

a journey of risk teaching concepts of risk within the mathematics classroom

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