Half-Full

Lesson Sketch: Probability - a matter of life and death

"In 1999 a woman in the UK, Sally Clark, was convicted of murdering her two children. They had both seemingly died from SIDS (more commonly known as cot death), an unexpected but natural cause of death.

One of the main pieces of evidence against her was the testimony of a Professor and expert on child-abuse. He used the probability of one child (in a family with no known factors that might increase the chance of SIDS) dying of SIDS (1 in 8543). He then squared this fraction to get the probability of BOTH children dying from SIDS being about 1 in 73 million.

The jury saw this number as the probability of an innocent explanation and, since it was so low, found her guilty of the murders. Unfortunately the professor who testified was not an expert on probability and miscalculated. The jury also did not understand the meaning of the statistic being calculated and drew the wrong conclusions from it.

In this series of lessons we will look at why this probability of 1 in 73 million is both wrong and irrelevant to the case. We will also look at how probability could have been properly used in this case."

This is the introduction to a series of lessons on probability.This case will frame the students' work on probability throughout the unit and not one student will ask "when am I ever going to use this?"

There are many options on how open or closed a project this is. Because I was short on time (being interviewed) I first asked them if they could spot any problems in the probability (useful for showing progression) before working in quite a closed manner, as a class, through section 1**.

Section 1 - Improbable events never happen

Introduce expected frequency of events. After students understand the general concept and are able to calculate the expected frequency of different events. Get them to work on the questions below:

The probability that both babies in a family of two will die of cot-death is 1/73'000'000.
There are about 340 million families with two children in the world.
1) How many families would you expect both babies to die of cot-death?

The probability of winning the lottery is 1/14'000'000.
About 7.5 million people play the lottery each week
2) How many winners would you expect to get:
i) Each week
ii) Each month
iii) Each year
iv) In 10 years

The probability of being born with 11 fingers or toes is 1/500.
3) What other information do you need to estimate the number of people in Bristol who were born with 11 fingers or toes?

4) Look at your answer to question 1. How is it related to the court case mentioned earlier? Does the answer to question 3 help you decide whether the woman was innocent or guilty?

5) Since the probability that both babies in a family of two will die naturally is 1/73'000'000, does that mean that the probability that both were murdered is 72'999'999/73'000'000? Explain why/why not?

I then got students to discuss question 4 in pairs before sharing with the group.
Concepts to get across:

• Probability is an estimate of frequency and whether something will happen or not depends a lot on the number of trials.
• Probability can not say for certain whether she is guilty or not as unlikely events do happen (though we can't be sure that's what happened here).
• The 73 million outcomes will include mostly families where neither child has died or a single child has died.

You could also go in to relative frequency here. This may help students understand where the 1 in 8543 statistic might have come from.

Section 2 - Dependent and Independent Events

Here you need to go in depth about the difference between dependent and independent events. Pupils need to be able to distinguish between the two, have some understanding about how the outcome of one event can change the probability of other events (e.g. if it rains today I am less likely to hang my washing out) and how to draw a probability tree from conditional probability problems.

What does this have to do with the case? Well, the probability of 1 in 8543 can only be squared if the death of the two children are independent events (otherwise the probability of the second death would be a different fraction). The causes of SIDS is not entirely known, but it is highly plausible that there could be genetic or environmental factors that would be common to the two children. This would mean that if there were a SIDS death in a family, then further SIDS deaths would be more likely than otherwise.

Students can work on these questions to see this in action*:

1a. The probability that a family's first baby will die of cot-death is 1/8543.
If the first baby dies of cot-death, the probability that the second will also die of cot-death is 1/442.
If the first baby DOES NOT die of cot-death, the probability that the second will die of cot-death is 1/11002.
Draw a probability tree to show this information.

b. What is the probability that:
 i. Neither child dies of cot-death?
 ii. One of the children die of cot-death?
 iii. Both children die of cot-death?

c. Using the number of families in the world with two children from the previous lesson (340 million), calculate how many families would you expect both babies to die of cot-death?

d. Why might the probability that the second child will die of cot-death be affected by whether the first child has also died of cot-death?

Section 3 - Further Conditional Probability

Now we get in to Bayes' theorem and the real reason why the 1 in 37 million statistic is simply irrelevant to the case:

As seen in question 5 from section one, the probability of 1 in 73'000'000, even if it were correct, is not the probability of innocence. It would instead be the probability that if you picked any random family of two children, both children would have died AND the cause of death was cot-death. This is not appropriate to use in this case because we already know that both children died.

Again, pupils could first work on standard conditional probability questions until they are comfortable with the function and usage of the above formula. Then pupils are going to calculate the probability that both children have died of cot-death GIVEN that both children have died. They can also compare this to the probability that both children were murdered GIVEN that both children have died:

The probability that both children in a family of two will be murdered is 1 in 10 million.
1. What is the probability that both children will die of cot-death OR murder (use the probability calculated in the previous session).

2. What is the probability that both children have died of cot-death given that both children have died of EITHER cot-death OR murder?

2. What is the probability that both children have been murdered given that both children have died of EITHER cot-death OR murder?

3. How do these two probabilities compare?

4. How does this change your view of the case?

5. Does this help you decide whether the woman was innocent or guilty? How?

Wrapping it up

A great way to recap all of this would be to get the students to prepare a letter to send to the judge of the case explaining the ways in which probability has been misused in this case and in what ways the calculations could be improved upon.
What is particularly great about this is that when they are done you can compare their letters to the letter written by the The Royal Statistical Society voicing their concerns (here) or the more specific letter written by Professor A.P. Dawid (of the R.S.S.) for her (successful) appeal (here)

Comments and Thoughts

*Although many of the figures used above are directly from the court case, and some are estimates based on other data I was able to find online, some of the data is currently unavailable and therefore is made up and should be used purely for illustration purposes.


**How I would prefer to run it:

• Start each class recapping previous work and ideas on the case
• Introduce the topic of the lesson and work on standard questions of that topic
• Ask students: "how this topic might be related to the case? What data would we need to apply today's skills to this case?"
• Give them the data they ask for and let them do the calculations.
• Ask students: "How does this new information change your view of the case? Does this help you decide whether the woman is innocent or guilty? Why?"

There are many other problems with the probability used in the case including (but definitely not limited to):

• Some environmental factors were included in the calculation, which make the probability of cot-death less likely, but many other factors in this individual case (which could increase the risk of cot-death) were not included.
• Cot-death is not the only alternative cause of death to the theorised double-murder, but the probability of these other causes of death were not included.
• The study where the 1 in 8543 figure came from was not intended for use in a criminal trial and instead is looking at possible causes of cot-death. It does not, therefore, directly apply to this case.

I thought for some time about the ethics of using a real, living person's tragedy as the basis for a Maths lesson. However, since this particular miscarriage of justice is due solely to the misunderstanding of basic probability by many of the involved parties, its use to encourage a better understanding of this topic seems appropriate. I wouldn't want to involve this in a quick and easy pseudo-context question. Instead it should be handled with the importance and gravitas it deserves.

Here are some other links about this case:

• Wikipedia
• UnderstandingUncertainty.org
• Berkeley University - This one is my favourite. Though it is more complex than some of the others, it has a lot more information about the lack of data in some areas and the assumptions that need to be made in order to make the calculations work. Interestingly, this paper argues that the probability of double-murder is much higher (over 99%) than stated in any of the letters endorsed by the R.S.S. (the point still stands, however, that statistics, if used in court at all, should be verified and reviewed by professionals).
• Bad Science

Interactive: Live Distance-Time Graph Creator

For a recent gifted and talented session I ran, I created This Excel File, which creates a distance time graph live. I gave pupils a pre-made distance time graph and pupils had to recreate it by walking across the room. I used it last week and it worked really great; pupils loved getting up and trying to time their movement. I'm sure it could be used in plenty of other ways too.

 

Setting up the room

• Make a clear path along the length of the room.
• Lay strips of tape in parallel lines spaced one meter apart across the length of the path.

Using the file

Open the file and click 'Enable Content' at the top.

Sheet1 - Creating the distance time graph

• Press reset to delete old values and set the view.
• The 'S' Column contains the timing of the 'Original' line. Change these for a different target graph. (Edit: Times have to be written in this format to work - hh:mm:ss)
• The 'T' Column contains the distance values that are shared between the 'Original' graph and the pupil graph. Change these if your room is a different size.
• Press prime to activate the distance time graph creator.
• Once primed, press enter (on the numeric keypad) every time the student steps across each line of tape, and a new point on the graph will created. (Note: first point created is at (0,0) and also starts the timer)

Sheet 2 - Working out speed
 

• 'View' sets the view
• Each button below this creates the triangles needed to calculate average speeds, but currently must be clicked in order (from top to bottom) for them to work. I will fix this eventually; at the moment the file is limited to my one use session, but I'd love to expand this to be a more multi-purpose graphing tool.

Lesson Sketch: A Deviation from Average Top Trumps *UPDATED*

Firstly, I have to admit that I haven't tried this one out yet. I just got the idea this weekend as I unearthed some 'fractions and decimals top trumps' I made for this Year 7 Maths Attack! I like the top trumps idea for a way to add a game element to practising maths skills (especially these on missing angles), but I also have two problems with them:

• To have enough for a whole class, you need a whole lot of printing, laminating and cutting. Either you do it or it eats in to precious class time.
• There's a low-limit to how hard you can make the problems before the time taken working out if you have won sucks all the fun out of playing.

Enter 'The Modal Group'; a set of Super-Average Super-Heroes!

Before you start, let the class decide on the four stats that you will be comparing (e.g. Strength, Wits, Speed, Etc.). Then give them these, nicely laid out in playing-card-sized rectangles:

Picture

Hero No. 1
Name:  .......................................................
Description:  .........................................................

Skills:
Constraints: Their four skills must have a mean of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 2

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a median of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 3

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a mode of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 4

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a range of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 5

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a median of 25 and a mean of 35.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 6

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a median of 35 and a mean of 25.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 7

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their first three skills must have a mean of 10, but all four skills must have a mean of 20 and a range of 50.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................


You could easily have more or make them easier/harder, but you get the idea. Pupils make their own Top Trumps cards and in the process:

• Get a feel for the differences between the mean, median, mode and range, seeing which ones are most affected by outliers and which average it is best to have higher (in this case).
• Solve more difficult problems involving averages, including finding the numbers when you know the averages, a moving average as data is added, and comparing data using averages.
• Get a reason to experiment and play around with the numbers and not just go for 40, 40, 40, 40 to get a mean of 40.
• Do something creative in class.
• End up with something that they can take home which will remind them of the lesson.

There you have it: Let pupils make the cards, checking their partner's to make sure they don't cheat, cut them out, and play against each other.

But wait, there's more! How about these for great follow-up questions:

• Which cards were the most/least constraining? Why?
• Which could you 'cheat' on? How?
• Next time I'll take those cards out to make it fair. Can you come up with some fairer constraints? Check that the problem you create is possible.
• What about a harder problem? What about an easier one?

As always, suggestions are suggested :)

*UPDATE* Just as I posted this I saw a related post on twitter by @InteractMaths. Would make a great follow-up or plenary to this lesson.

Lesson Sketch: Relative frequency - A lesson on calculating the risks of sport

Another basic one here. Why do it? Because you can't repeat the 'checking to see if a dice is fair' activity year on year forever. Step one - Ask pupils to put a list of common sports in to the order of how dangerous they think the sports are:

American Football, Football, Gymnastics, Ice Hockey, Swimming, Wrestling, Field Hockey, Skiing, Water Polo, Equestrian, Golf, Skydiving

Step 2 - Show the number of deaths for each sport and proclaim American Football to be the most dangerous!

Sports	Deaths in US colleges in the last 5 years in five years
American Football	55
Football	6
Gymnastics	1
Ice Hockey	1
Swimming	8
Wrestling	3
Field Hockey	0
Skiing	2
Water Polo	1
Equestrian	1
Golf	0

Ask what is wrong with your statement. Why have so many more people died in American Football than any of the others? Get that we also need to know how many people actually took part. Show them the third column of the table along with the formula for relative frequency:

Sports

Deaths in colleges

in five years

Participation in colleges

in five years

American Football

55

2’079’069

Football

6

841’963

Gymnastics

1

57’613

Ice Hockey

1

128’898

Swimming

8

475’953

Wrestling

3

188’272

Field Hockey

0

156’370

Skiing

2

33’974

Water Polo

1

39’968

Equestrian

1

10’489

Golf

0

62’528

Pupils put the sports in order of their danger of death. One thing I like about this is that the data (got from here) is not really adequate for most of them and it gives plenty of room for discussion.


Step 3 - Show pupils this pdf: http://www.uspa.org/Portals/0/MembershipSurveys/memsurvey09.pdf

Ask how we can work out the relative frequency of deaths in skydiving using this data. Pupils will probably go for the 16 deaths over 2.5 million jumps, but for the result to be comparable to the previous task it would make more sense to do the number of deaths over the 32'177 members. This is because all the other sports will be played throughout the year and it is counted as deaths per participant rather than deaths per game. An interesting and fairly advanced conversation to be having with your pupils - YAY!

Gamification of the Classroom?

It seems that lately I'm hearing about gamification (applying the mechanics of gaming) of everything, including education. My initial reaction was wary, since gaming and education have different core objectives (entertainment and learning respectively). However, reading this article on 'Cognitive Flow: The Psychology of Great Game Design', I couldn't help but think of the similarities between its discussion of feedback and my ideas of what makes good assessment.

This has inspired me to think about gamification and what features of game design it may be useful to apply to teaching. This is just an open thought experiment, so if you have anything to add, or disagree with, please let me know. Note that as well as the article mentioned above, I will also reference "Creating Flow, Motivation & Fun in Learning Games".

Characteristic 1: Games should have concrete goals with manageable rules.

This seems a fairly obvious statement when applied to the classroom. Pupils are generally happier when they know what is expected of them and its why there is so much emphasis from OFSTED about making learning objectives (L.O.s) and outcomes clear throughout the lesson. Ways to achieve this:

• Make the goals plainly comprehensible in every part of the lesson. This could include linking explanations, questions and assessments to the L.O.s. Particularly useful may be the questions, so that 'if you can do these questions then you have met this L.O.'.
• Goals and new skills need to be introduced with a minimum of distractions.
• Time should be taken to train the pupil on new skills in a low risk environment (e.g. when you get the Gravity Gun in Half-Life you first use it to play fetch with dog). Traditionally this is achieved by examples and practice questions. You could also use group activities where pupils can confirm each other's correct answers.
• 'The completion of small goals (e.g., clearing a field of boars) links to larger goals (e.g., getting enough XP to level up), which in turn link to even larger goals (e.g., getting access to level-specific gear). This linkage creates a series of rewarding experiences that can hook gamers to a game and create the goal-achievement-reward cycle.' This link is certainly present in education; learning skills, connecting with other skills, enabling a higher test score etc., which all links to your final exams. I do think though that this link is not always clear. Sharing standards achieved and how it links to termly reports etc. (see SBG) would be one way of doing this. Another link which I will talk about more below is the link between classwork and 'maths in the real world'.

Characteristic 2: Games should only demand actions that fit within a player's capabilities.

This is obviously desirable, particularly with pupils who lack confidence, but is not always easy to acheive in a class of 30+. Use of optional support and extension tasks can help to find the right difficulty, as can rich tasks, which allow for multiple entry points and investigation. This is similar to the game design idea of Dynamic Difficulty Adjustment.

Other things mentioned are how games should 'avoid introducing a lot of skills at the same time', 'avoid the tendency to over-specify', sticking only to the most important parts, and 'start out simple, with minimal information, and add in new data as needed'.

Characteristic 3: Games should give clear and timely feedback on player performance.

This is what initially caught my attention as a lot of SBG is about creating the opportunity for timely and useful feedback, and in fact people have already discussed its similarities to game systems. Its also what I like about Kate Nowak's Row games, because pupils get almost instant feedback on each question. Its also something that computer apps, such as this excellent algebraic equations app, are capable of achieving.

This section also mentions setting up feedback systems from the start and maintaining them. This is something I've had trouble with in the past and am currently trying to focus on being more consistent with merits, etc.

Characteristic 4: Games should remove any extraneous information that inhibits concentration.

This links to Dan Meyer's discussion of pseudocontext, where he invites you to ask whether the context adds anything to the problem or just disguises it.

Getting people to repeat tasks.

Repetition is obviously conducive to learning, but its sometimes hard to get pupils to perservere when they don't immediately understand. Here are some ways that games can do it:

• Most games simply let players 'try again' without any stigma attached. This is harder to achieve in a classroom, but it probably starts with encouraging a "growth mindset" culture that embraces initial difficulties, like this 'favourite no' idea.
• 'Give the player enough feedback so they can figure out how to improve their performance the next time.' This can be achieved through direct instruction and questioning or group/peer work. Its also another stated aim of SBG.
• 'Allow players to skip excessive and meaningless repetition of the same skill. Focus on skills related to the learning objectives, let the player know when they succeeded, and move on.' Having a harder task that pupils can move on to when they feel they've mastered the basics. Dan's version of SBG has pupils skip problems they already have evidence of mastery with. This is something that I really feel needs improvement in schools. For example, a popular answer to the question 'When Will I Ever Use This????' seems to be that pupils should treat math class like brain training, like how a footballer does weights. However, where is the motivation if you never actually get them to play football? The gaming equivalent would be a computer game that was just one long tutorial. Functional maths tasks and Dan Meyer's 3Acts/WCYDWT problems are a couple of ways to address this. Here, you are applying maths in unmistakably real-world situations. They may be hand picked to be simple enough for pupils to understand, but they do not feel contrived and because they have multiple entry points and solution strategies, they allow pupils to 'play' with the skills they have learned.
  
• 'Make the repeated task feel different each time around. This means providing choices, actions, and control so that the player can become engaged in a similar but slightly different experience. Alternately, the next time around, the process should go much faster. This allows them to enjoy the experience of mastery over previously challenging content.' This one has stumped me. Though it sounds entirely applicable to education, I can't really think of ways to do it. Any help on this?

Lesson Sketch: Land speed

Hopefully I'll get back into blogging soon, but for now here's a quick puzzle about converting units:

Have a guess: which of these animals is fastest/second fastest/slowest?
Using the measurements given, put the animals in order from fastest to slowest.

Cheetah - 70 miles per hour

Snake - 9 metres per second

Human - 751 metres per minute

Grizzly Bear - 1159 kilometres per day

Reindeer - 52 kilometres per hour

Elephant - Can travel 1 kilometre in 89 seconds

How much faster than Ussain Bolt could a cheetah run the 100m?

Answer:

Lesson Sketch: Using Biometrics to Run like Usain Bolt

http://www.fotopedia.com/items/josegoulao-TbuZScZeYnM

Intro:
Students get into pairs with a stop-watch each. The pairs time each other running 100m, getting split times for every 10m (you will need to have marked the track for this in advance.

Main:
Say we will now try to improve those times by learning from the best: Usain Bolt. Show the pupils graphs of Usain Bolt's 200m World Record run:

 



Students draw a distance/time (and speed/time if pupils have learned about tangents) of their own split times.

1. How is your graph different to Usain's (apart from his being a 200m run)?
2. Using Usain's speed/time graph, describe his actions in a much detail as possible. Give times for each action. Give distances for each action.
3. Why do you think Usain slows down towards the end?
4. Compare Usain's acceleration at the start to yours. What other differences are there between Usain's graphs and yours?

Hopefully from that, students will see that Usain accelerated extremely quickly in the first 10 meters, but continued to accelerate until about the 50 meter mark. From there he tried to maintain his top speed, but slowed down gradually.

(Optional): Show a video of this run. Note techniques: Arms rigid with fast pumping movement, shortish steps, leaning forward at start but upright after a few seconds.

End:
Write down a checklist of what Usain did in the run. Pupils go back to the running track in their pairs and try to do everything on the checklist to improve their previous time.