Friday, 6 June 2014

The False-Positive Paradox as a Class Activity/Discussion Point

There is a test that screens for some rare disease (only 1 in every million people is infected). It is 99% accurate (it gives the correct answer of positive or negative 99% of the time). You go for the test and to your dismay it returns a positive result!
What is the chance that you actually have the disease?

You may think that its pretty likely you have the disease. However, from the information above, the actual probability that you have the disease is still very unlikely (1 in 10'102). This is the basic premise of the false-positive paradox.

I don't really like that it is named as a paradox, because it is not actually in any way paradoxical, just counter-intuitive. It works because there are so many more non-infected people (999'999) that each have a 1% chance of getting a false positive than the number of infected people (1) who has a 99% chance of getting a true positive result.



This counter-intuition can cause exactly the sort of cognitive-disonance that can be so useful in getting people to realise that their current understanding needs improving, but in its current form its pretty wordy, inaccessible and abstract. I wanted to make it in to a class activity. Instructions below.

 

 

 

How to run the Class Activity/Discussion Point:

  • As students enter the room, give them each a slip of paper with a random word on it. Tell them not to show anyone else.
  • Once they're settled, tell them that there is a (make-believe) disease going round school that only affects about one student per class. In this class the one student, the one with the word banana on their slip of paper, is infected (Don't tell anyone if its you or not!).
  • Tell them you have a way of checking if anyone has the disease, but its not always accurate. To perform the test, you will ask the student if they are infected. Before answering, the student will roll a dice and will tell the truth if they roll a 1, 2, 3, 4 or 5, but they will lie if they roll a 6.
  • Perform this test on some students until someone claims to be infected.
  • Ask the class how likely is it that this student is actually infected. Ask for justifications. Most will say very likely, some may evan say 5/6.
  • Perform the test on a few more students until you get a few more claiming to be infected.
  • But they can't all be 'very likely'! What is going on here? 

By the way, make sure the students know the disease is not real!


Would this work in your classroom? Could it be improved? At the moment its just an idea and I'd like to know what you think.

Tuesday, 18 March 2014

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

Thursday, 19 December 2013

The True Cost of Gaming: Adjusting for Inflation

I was inspired by this image:
by Auir2blaze

If I was American, I would just blank out some of the numbers and call it an activity. Being from the UK, I had to do a bit of leg-work:
There are plenty of intriguing questions here, such as:
  • Which Xbox was cheapest?
  • Which is the most expensive console ever?
  • Are consoles cheaper to buy in the US or the UK? (using the original image as a reference and a conversion rate taken from the internet)
  • How can we have percentages that are more than 100%?
  • What did SEGA do wrong? How did it lose the console war? (Speculation)

Answers:

Link to table on a google doc

Thursday, 12 December 2013

Improving Functional Skills in Mathematics

As mentioned in a previous post, I have been thinking lately about how to attack Functional Skills Mathematics to improve results. I have decided on a 4-pronged approach.

Prong 1 - Improving literacy in maths

I have already made a post with ideas and resources for this, which you can find by clicking here.


Prong 2 - Improving problem-solving skills

As functional skills is so focussed on long, involved questions. Students need to feel confortable deciding what steps they need to take and set out their work so that they can follow it and check for mistakes.
  • 3-Act Maths - I find this way of working to be an ideal way to introduce problem-solving to students in a non-threatening way. Starting with just the actual question and no constraints, variables or other information means that problems don't even look mathematical at first and students feel free to take guesses, make assumptions and bring in outside knowledge. Getting students to work out 'the information they would like in order to work out the answer' means that, when you bring those variables in, they are no longer confusing and unclear, but are now the very tools the students were asking for. I've used 3Acts from Dan Meyer's list, problems over at Finding Ways to Nguyen, and picture and video prompts from 101qs. I also sometimes just take a book question, skip all the set-up to the question and just ask the actual question (e.g. How long would it take me to cycle to Edinburgh? Can we get enough rain-water from the college roof to have toilets that run on the recycled water? etc.).
  • Structuring of workings out - e.g. using the problem scaffolder from the previous post on literacy. The idea of this is to help students to think clearly about the steps they are making and what they are trying to calculcate from each step.
  • Discussion of problem-solving strategies - I made some posters last year which I have up on the wall. Before tackling a problem I ask "which strategy could we use here? Why will that help?", when students are stuck I ask "which strategies have you tried? which might you try next?", when they finish I ask "which strategies did you use? did anybody use a different strategy?" Getting students to a point where they're comfortable with using these strategies and are starting to get a feel for when each one is applicable can only help their confidence when faced with a difficult problem.
  • Problem Posing - I originally got this idea from Matt Ives (blog post about it here). Basically the idea is to move focus away from getting the answer and move it on to thinking about the steps you would take and what information you would need. Matt Ives' way of achieving this: don't have any numbers, so pupils can't work out the answer, instead rewarding students who work out the steps they would hypothetically take to find the solution. He has a whole raft of them here on scribd and they've worked very well in the past, once students get away from trying to find an answer.

Prong 3 - Improving mathematical skills

  • Regular lessons - In amongst the literacy and problem-focused lessons, I will also include regular skills lessons. These will come before more complicated problems that involve these skills (along with others) so that students get to see them in context.
  • Starters and Homework - I will also include general basic skill revision in lesson starters and homeworks.
  • Fill in the blanks on calculation tables are a great way to make repetitive practice of a skill more interesting. I wrote a post about it here.

Prong 4 - Putting it together (and improving clear working)

  • Exam questions - The previous prongs will then be put in to practice when we do past-paper questions in class and for homework. I will link the problems to the work we have done on literacy and problem solving and discuss the skills that will be needed to solve the question.
  • 3-Act questions rewritten as past paper questions and vice-versa - After solving a 3-Act problem, I will show a version of how I think it would look like in an exam paper. Hopefully this should make students feel empowered that they've just solved such a difficult-looking question and will help them see the link between the problem-based lessons and the actual exams. After practicing an exam problem I will get students to try to imagine how it would look as a 3-act problem. Not only does this reinforce the link between the two, but help students decide which part is the question, the constraints, the variables, etc.
  • Peer marking - Will help students to see the importance of clear and methodical working as well as to think about different problem-solving strategies that others have used in their working.

Secret prong 5 - Improving Engagement

Overarching all of this is what I feel that I actually do best as a teacher, which is to show students that maths can be enjoyable and engaging. I achieve this in the following ways:
  • Improving accessibility - 3-Act maths are great for this, as are other rich tasks, and the fill-in-the-blanks calculation tables. This is because of the multiple entry points to the questions, and the lack of immediate reading and comprehension required. If students' literacy can be improved, then the more wordy questions will become less and less daunting.
  • Improving confidence - As students get more and more comfortable with multi-step problems they will see that there are things that they can achieve in maths. This never becomes obvious to students who only practice the basic, underlying skills, no matter how good they get at them.
  • Improving relevancy - By this I don't necessarily mean 'real' maths, but I mean maths that students can imagine themselves using in their lives. This is where are a focus on problem-solving strategies can be helpful, because even though the problem itself might not relate strongly to them, being able to use the strategies is applicable by anyone.
  • The strength of my sparkling personality? ...
  

Friday, 6 December 2013

Inspiring Mathematical Applications: Randall Munroe answers all

As a Maths teacher, I'm guessing you'll already be familiar with Randall's awesomely nerdy web-comic, xkcd:

Maybe you're also aware of his "What If?" site. If you're not, I'd like to introduce you to it. Each week Randall Munroe, a physicist and former NASA employee, answers question that people post to him online. Though the questions are usually strange ("How fast can you hit a speed bump while driving and live?") and sometimes ridiculous ("If you suddenly began rising steadily at one foot per second, how exactly would you die? Would you freeze or suffocate first? Or something else?"), Randall tries to answer them as accurately as he can, backing up his statements with citations and calculations.

Though, the maths sometimes gets quite complex, the funny topics and humourous pictures combine to make it much more approachable than it otherwise would be.

I love to read them just for myself, but I've also started bringing them in to some of my classes. Last week I found lots of ideas and resources for improving literacy in maths. This week, one of the ideas I tried was about scaffolding the problem, specifically splitting a question in to its important parts (The actual question, the information given, the constraints, what calculations are needed, etc.). As a plenary I asked them to guess the answer to, "Is it possible to build a jetpack using downward firing machine guns?", then to read through the What If? article about it, whilst highlighting the different components of the text (The question, the information gathered from research, the calculations, the constraints, etc.).

It worked very well, and the students really saw the link between the work they'd just done and the work done in the article. I'm hoping that this link will help them to feel empowered (because the techniques they are using work on much more complicated questions) rather than disenchanted by the complexity of the maths.

Tuesday, 3 December 2013

Improving Literacy in Maths

I've been thinking, lately, about how to improve literacy in Maths. If you would like to read the strategies I have so far come up with, skip to the first heading.

For the first time this year I am teaching Functional Skills Mathematics (Levels 1 and 2). The scheme of work I have for these classes is pretty bare-bones and is very skills-based. I am finding it hard to stick straight to the scheme of work, as most of these skills are actually more basic than what my students have already seen when attempting their GCSE and are made even simpler as there is no non-calculator test.

Historically, many students fail the level 2 course and when I asked my colleagues why, they all said that it was down to poor literacy skills. This is understandable when the questions all look like this:

 


Obviously literacy isn't the only difficulty with this question, so I have decided to teach Functional Skills with a four-pronged approach. I will go in to detail about the other prongs of my plan in a separate post... Prongs.


Prong one: Ways to Improve Literacy

1. Scaffolding questions and correct working

This can be done in different ways.
  • So far I have been getting them to write titles for each seperate part of their working before starting any calculations, so that they can then split the work in to more managable chunks.
  • In future I will try using a problem scafforlding sheet like this:
  • Problems with clue cards would also help to split up a question and promote discussion and team-work. It works like this: Each group gets a simple question. Each individual in the group also gets a different card with a single piece of information that is needed to answer the main question. Members of the group have to decide how to use and combine the information sheets they are given. Here is an example (For lots more, click here):
  • Question sequencing: All the correct working for a set of problems is written down on separate tabs of paper, but their order has been mixed up. Students must put the working in to a correct order.


2. Building vocabulary and making connections

This could include:
  • Word-walls: Where mathematical words can be written with their definitions and/or diagrams and grouped according to connections with other words.
  • Connect-two: You have a list of words at the top (eg. Percentage, fraction, TAX, Chocolate). You pick different pairs of them and explain in what way they are connected and give examples.
  • Word-splash: Similar to connect-two, but the words are on the board and you pick people to make a connection.
  • Anticipation guide: A table where the first column consists of statments and/or questions (eg. A decimal number is a number less than 1). The second and third columns give a space where students decide whether they think the statement is true or false and why. The second column gets their opinions before working on the subject and the third column gets it after working on the subject (good evidence of progression here too!).
  • Tarsias: These are jigsaws where students match a question to an answer (or a word to its definition). Mr Barton has loads available here. You could also print off a blank tarsia and get students to create one.
  • 3 facts and a fib: Students write down three facts and a lie about a maths object, word or topic. They try to make the lie believable enough that when they pass it to their neighbour, the neighbour can not work out which statement is the lie.
  • Definition map: to the example below I would add a space for illustrations of the word or concept:
  • Taboo: The classic game game be adapted easily to a nice plenary activity - give them a maths password (eg. Mutually-exclusive) and a list of taboo words that they are not allowed to use (eg. Probability, events, same time, both). The chosen student then has to describe the password, without using any taboo words, to the class until someone can guess it.
  • Pictionary: Similar to taboo except that you make drawings to represent the word.
  • The Mathematics Assessment Project has many matching activities designed to promote discussions and expose misconceptions, as well as Professional Development modules to help you use them effectively. Some of them are excellent.
  • Think, pair, share: Students work on their own on a question. In this time they may do some calculations but should focus more on organising the information and making notes. Next they get in to pairs. First they should take it in turns explaining their thoughts (one talker, one listener) so far, before working together to answer the question. Finally, they join another group and explain how they got to their answer (again taking it in turns so that both groups get to explain their work).
  • Odd one out: Take three different mathematical objects (eg. triangles) and get students to decide which is the odd one out, then discuss their decision with their partner. This works best if each object could be the odd one out for a different reason (isoceles, right-angled, different area, different perimeter, etc.). That way different students get a different answer and their discussions lead others to see the objects in another way.
  • Carroll diagrams: Two-way tables where students have to sort objects in to the correct section that matches the description of that word. Here is a good example.


3. Peer marking and constructive discussion

  • Marking each others' work will help them understand the improtance of clear working, with titles, explanations,etc. Discussing how a correct answer could be improved to make it more clear would also help with this.
  • Always, sometimes, never true: Students read a statement and decide whether it is always true, never true, or sometimes true. They then justify their answer and compare results with their peers (examples can be found here).
  • Rally coaching -  Students work in pairs. They take it in turns with one person answering a single question, whilst the other coaches. Coaching could simply be helping to solve the questions or could be more complex, with rules like, "avoid telling them what to do. Try to ask a question that will help them to figure out the next step."
  • Many of the activities from section 2 would also apply here.
  • Bowland Maths have a Professional Devolopment activity (module 3) described as "Fostering and Managing Collaborative Work: How can I get them to stop talking and start discussing?" I have not been able to look at this yet  as there is no sound on my computer at school, but I will update the page when I do. I have already enjoyed and would recommend some of the other PD modules on this site though.


4. Filtering information from text

  • Students highlight the different parts of a question in different colours (question, given figures, constraints, etc.)
  • Students write the gist of a block of text using only 20 words. Students could create an individual version, then a version in pairs, before writing a final one in their books.
  • There is a video series (click here) with worksheets (click here for the worksheets), where you are asked to fill in information as you hear it on the videos.


Credits

Much of these ideas (and plenty more ideas!) have come from:
  • "Growing the Connection Between Mathematics and Best-Practice Reading and Writing Strategies" by Jennifer Kosiak, Sue Schumann, Ann Harry and Bonnie Jancik (available here)
  • "WHAT’S LITERACY GOT TO DO WITH IT? Literacy in the Math and Science Classroom" by Blair Covino and Barb Mazzolini  (available here)
  • "Literacy in Maths" by The Highland Curriculum for Excellence (available here)
  • Other sources that have been linked above: Cumbria Grid for Learning, Mr Barton, Mathematics Assessment Project, Nrich, Bowland Maths and Channel 4 Learning.